| L(s) = 1 | − 5·4-s + 6·5-s + 12·9-s + 6·11-s + 15·16-s − 8·19-s − 30·20-s + 20·25-s − 22·29-s + 46·31-s − 60·36-s − 14·41-s − 30·44-s + 72·45-s + 5·49-s + 36·55-s − 40·59-s − 18·61-s − 35·64-s + 12·71-s + 40·76-s − 40·79-s + 90·80-s + 64·81-s − 24·89-s − 48·95-s + 72·99-s + ⋯ |
| L(s) = 1 | − 5/2·4-s + 2.68·5-s + 4·9-s + 1.80·11-s + 15/4·16-s − 1.83·19-s − 6.70·20-s + 4·25-s − 4.08·29-s + 8.26·31-s − 10·36-s − 2.18·41-s − 4.52·44-s + 10.7·45-s + 5/7·49-s + 4.85·55-s − 5.20·59-s − 2.30·61-s − 4.37·64-s + 1.42·71-s + 4.58·76-s − 4.50·79-s + 10.0·80-s + 64/9·81-s − 2.54·89-s − 4.92·95-s + 7.23·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.545498154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.545498154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{5} \) |
| 5 | \( 1 - 6 T + 16 T^{2} - 34 T^{3} + 24 p T^{4} - 14 p^{2} T^{5} + 24 p^{2} T^{6} - 34 p^{2} T^{7} + 16 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 37 | \( ( 1 + T^{2} )^{5} \) |
| good | 3 | \( 1 - 4 p T^{2} + 80 T^{4} - 44 p^{2} T^{6} + 1600 T^{8} - 5318 T^{10} + 1600 p^{2} T^{12} - 44 p^{6} T^{14} + 80 p^{6} T^{16} - 4 p^{9} T^{18} + p^{10} T^{20} \) |
| 7 | \( 1 - 5 T^{2} + 157 T^{4} - 748 T^{6} + 11922 T^{8} - 50910 T^{10} + 11922 p^{2} T^{12} - 748 p^{4} T^{14} + 157 p^{6} T^{16} - 5 p^{8} T^{18} + p^{10} T^{20} \) |
| 11 | \( ( 1 - 3 T + 32 T^{2} - 61 T^{3} + 470 T^{4} - 617 T^{5} + 470 p T^{6} - 61 p^{2} T^{7} + 32 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 13 | \( 1 - 36 T^{2} + 456 T^{4} - 1724 T^{6} - 49056 T^{8} + 1168374 T^{10} - 49056 p^{2} T^{12} - 1724 p^{4} T^{14} + 456 p^{6} T^{16} - 36 p^{8} T^{18} + p^{10} T^{20} \) |
| 17 | \( 1 - 81 T^{2} + 3653 T^{4} - 114652 T^{6} + 2746906 T^{8} - 52194214 T^{10} + 2746906 p^{2} T^{12} - 114652 p^{4} T^{14} + 3653 p^{6} T^{16} - 81 p^{8} T^{18} + p^{10} T^{20} \) |
| 19 | \( ( 1 + 4 T + 55 T^{2} + 128 T^{3} + 1410 T^{4} + 2616 T^{5} + 1410 p T^{6} + 128 p^{2} T^{7} + 55 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 23 | \( 1 - 144 T^{2} + 10280 T^{4} - 482952 T^{6} + 16597040 T^{8} - 434849678 T^{10} + 16597040 p^{2} T^{12} - 482952 p^{4} T^{14} + 10280 p^{6} T^{16} - 144 p^{8} T^{18} + p^{10} T^{20} \) |
| 29 | \( ( 1 + 11 T + 136 T^{2} + 1025 T^{3} + 7632 T^{4} + 41283 T^{5} + 7632 p T^{6} + 1025 p^{2} T^{7} + 136 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 31 | \( ( 1 - 23 T + 262 T^{2} - 2197 T^{3} + 16362 T^{4} - 102519 T^{5} + 16362 p T^{6} - 2197 p^{2} T^{7} + 262 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 41 | \( ( 1 + 7 T + 86 T^{2} - 3 T^{3} + 796 T^{4} - 17959 T^{5} + 796 p T^{6} - 3 p^{2} T^{7} + 86 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 43 | \( 1 - 245 T^{2} + 30581 T^{4} - 2525036 T^{6} + 154591042 T^{8} - 7429518878 T^{10} + 154591042 p^{2} T^{12} - 2525036 p^{4} T^{14} + 30581 p^{6} T^{16} - 245 p^{8} T^{18} + p^{10} T^{20} \) |
| 47 | \( 1 - 238 T^{2} + 30269 T^{4} - 2575368 T^{6} + 166157154 T^{8} - 8597995284 T^{10} + 166157154 p^{2} T^{12} - 2575368 p^{4} T^{14} + 30269 p^{6} T^{16} - 238 p^{8} T^{18} + p^{10} T^{20} \) |
| 53 | \( 1 - 153 T^{2} + 19773 T^{4} - 1677340 T^{6} + 124372154 T^{8} - 6992655990 T^{10} + 124372154 p^{2} T^{12} - 1677340 p^{4} T^{14} + 19773 p^{6} T^{16} - 153 p^{8} T^{18} + p^{10} T^{20} \) |
| 59 | \( ( 1 + 20 T + 347 T^{2} + 4048 T^{3} + 41662 T^{4} + 341560 T^{5} + 41662 p T^{6} + 4048 p^{2} T^{7} + 347 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 61 | \( ( 1 + 9 T + 224 T^{2} + 1203 T^{3} + 19448 T^{4} + 77465 T^{5} + 19448 p T^{6} + 1203 p^{2} T^{7} + 224 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 67 | \( 1 - 468 T^{2} + 101704 T^{4} - 13829796 T^{6} + 1345998088 T^{8} - 101216082902 T^{10} + 1345998088 p^{2} T^{12} - 13829796 p^{4} T^{14} + 101704 p^{6} T^{16} - 468 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( ( 1 - 6 T + 171 T^{2} - 792 T^{3} + 15010 T^{4} - 65764 T^{5} + 15010 p T^{6} - 792 p^{2} T^{7} + 171 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 73 | \( 1 - 300 T^{2} + 51832 T^{4} - 6366948 T^{6} + 615079552 T^{8} - 49160084606 T^{10} + 615079552 p^{2} T^{12} - 6366948 p^{4} T^{14} + 51832 p^{6} T^{16} - 300 p^{8} T^{18} + p^{10} T^{20} \) |
| 79 | \( ( 1 + 20 T + 382 T^{2} + 4700 T^{3} + 54438 T^{4} + 501128 T^{5} + 54438 p T^{6} + 4700 p^{2} T^{7} + 382 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( 1 - 446 T^{2} + 102229 T^{4} - 16080104 T^{6} + 1905380754 T^{8} - 177185644148 T^{10} + 1905380754 p^{2} T^{12} - 16080104 p^{4} T^{14} + 102229 p^{6} T^{16} - 446 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( ( 1 + 12 T + 301 T^{2} + 2480 T^{3} + 44794 T^{4} + 306888 T^{5} + 44794 p T^{6} + 2480 p^{2} T^{7} + 301 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 97 | \( 1 - 473 T^{2} + 109701 T^{4} - 17966844 T^{6} + 2393147002 T^{8} - 259837694166 T^{10} + 2393147002 p^{2} T^{12} - 17966844 p^{4} T^{14} + 109701 p^{6} T^{16} - 473 p^{8} T^{18} + p^{10} T^{20} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.39486599754975174503097917282, −4.25157443192212301060702056408, −4.18685234094914153217461541719, −4.15397684502326706821031253525, −4.06435275898140240826563018683, −3.88591168176641197008988604593, −3.79464750291226696565121795316, −3.49608328756568861810200040506, −3.28305222794189869118810119477, −3.20060964081897222863812785550, −3.03447413166692801916792642754, −2.94838572005991507048950569191, −2.77829908835786729281623290393, −2.68946954117892575433683834986, −2.56422612539408917833572604751, −2.12798534705374300661897601621, −1.92885069085282229420940045713, −1.84213289369450174461901842888, −1.67255219193717623295780646184, −1.53160559767277343429938742611, −1.43742774170773838884226522745, −1.36728266841516739548101221167, −1.15742013890095378258044901024, −0.789106106064465189110723694796, −0.42785923145299655776089429301,
0.42785923145299655776089429301, 0.789106106064465189110723694796, 1.15742013890095378258044901024, 1.36728266841516739548101221167, 1.43742774170773838884226522745, 1.53160559767277343429938742611, 1.67255219193717623295780646184, 1.84213289369450174461901842888, 1.92885069085282229420940045713, 2.12798534705374300661897601621, 2.56422612539408917833572604751, 2.68946954117892575433683834986, 2.77829908835786729281623290393, 2.94838572005991507048950569191, 3.03447413166692801916792642754, 3.20060964081897222863812785550, 3.28305222794189869118810119477, 3.49608328756568861810200040506, 3.79464750291226696565121795316, 3.88591168176641197008988604593, 4.06435275898140240826563018683, 4.15397684502326706821031253525, 4.18685234094914153217461541719, 4.25157443192212301060702056408, 4.39486599754975174503097917282
Plot not available for L-functions of degree greater than 10.
