| L(s) = 1 | + 2·5-s − 3·9-s + 6·11-s − 4·19-s + 25-s + 16·29-s − 16·31-s − 16·41-s − 6·45-s + 10·49-s + 12·55-s − 16·59-s + 4·61-s + 24·71-s − 12·79-s + 6·81-s − 4·89-s − 8·95-s − 18·99-s − 80·101-s + 12·109-s + 21·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s + 1.80·11-s − 0.917·19-s + 1/5·25-s + 2.97·29-s − 2.87·31-s − 2.49·41-s − 0.894·45-s + 10/7·49-s + 1.61·55-s − 2.08·59-s + 0.512·61-s + 2.84·71-s − 1.35·79-s + 2/3·81-s − 0.423·89-s − 0.820·95-s − 1.80·99-s − 7.96·101-s + 1.14·109-s + 1.90·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.564574126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.564574126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | \( ( 1 - T )^{6} \) |
| good | 7 | \( 1 - 10 T^{2} + 31 T^{4} - 124 T^{6} + 31 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 + 14 T^{2} + 311 T^{4} + 4692 T^{6} + 311 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 30 T^{2} + 1151 T^{4} - 18164 T^{6} + 1151 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 2 T + 5 T^{2} - 108 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{3} \) |
| 29 | \( ( 1 - 8 T + 3 p T^{2} - 432 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 8 T + 101 T^{2} + 480 T^{3} + 101 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 174 T^{2} + 13943 T^{4} - 655652 T^{6} + 13943 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 8 T + 3 p T^{2} + 624 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 226 T^{2} + 22423 T^{4} - 1251628 T^{6} + 22423 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 186 T^{2} + 17903 T^{4} - 1043180 T^{6} + 17903 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 190 T^{2} + 20375 T^{4} - 1316100 T^{6} + 20375 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 2 T + 131 T^{2} - 204 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 226 T^{2} + 28103 T^{4} - 2235900 T^{6} + 28103 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 12 T + 245 T^{2} - 1688 T^{3} + 245 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 234 T^{2} + 27279 T^{4} - 2258332 T^{6} + 27279 p^{2} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 6 T + 233 T^{2} + 940 T^{3} + 233 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 486 T^{2} + 99383 T^{4} - 10945028 T^{6} + 99383 p^{2} T^{8} - 486 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 2 T + 255 T^{2} + 348 T^{3} + 255 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 454 T^{2} + 94543 T^{4} - 11606932 T^{6} + 94543 p^{2} T^{8} - 454 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.46594944128980453040777266346, −4.42885752643919455481606105279, −4.39892290537778238486795716724, −4.19774384854639048745462260440, −3.90567108098387751749639048552, −3.86091782906617814168235339385, −3.65507327557211832922316719412, −3.52303173437102395124376152919, −3.50543622828446913261807195992, −3.25956660427373176245916677486, −2.97215898817102099861262058599, −2.90823864962864757402690948332, −2.77514270505209182235749349670, −2.42771267176284252716214710961, −2.40387526881758131931601682775, −2.39371914054203298518144493601, −1.91266082906115039153081005636, −1.77532473541198652605608151932, −1.72359574976105642656954148587, −1.46287185118712345549711804443, −1.28855809257500958653215153678, −1.10955926487437224064424706370, −0.841842234192493177452283137314, −0.34006363423594707186063484846, −0.27579647668228536578355633447,
0.27579647668228536578355633447, 0.34006363423594707186063484846, 0.841842234192493177452283137314, 1.10955926487437224064424706370, 1.28855809257500958653215153678, 1.46287185118712345549711804443, 1.72359574976105642656954148587, 1.77532473541198652605608151932, 1.91266082906115039153081005636, 2.39371914054203298518144493601, 2.40387526881758131931601682775, 2.42771267176284252716214710961, 2.77514270505209182235749349670, 2.90823864962864757402690948332, 2.97215898817102099861262058599, 3.25956660427373176245916677486, 3.50543622828446913261807195992, 3.52303173437102395124376152919, 3.65507327557211832922316719412, 3.86091782906617814168235339385, 3.90567108098387751749639048552, 4.19774384854639048745462260440, 4.39892290537778238486795716724, 4.42885752643919455481606105279, 4.46594944128980453040777266346
Plot not available for L-functions of degree greater than 10.
