| L(s) = 1 | + 6·3-s + 5·4-s + 21·9-s + 30·12-s − 2·13-s + 10·16-s + 4·23-s + 10·25-s + 56·27-s − 4·29-s + 105·36-s − 12·39-s − 32·43-s + 60·48-s − 3·49-s − 10·52-s − 36·53-s + 28·61-s + 10·64-s + 24·69-s + 60·75-s − 32·79-s + 126·81-s − 24·87-s + 20·92-s + 50·100-s − 24·101-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 5/2·4-s + 7·9-s + 8.66·12-s − 0.554·13-s + 5/2·16-s + 0.834·23-s + 2·25-s + 10.7·27-s − 0.742·29-s + 35/2·36-s − 1.92·39-s − 4.87·43-s + 8.66·48-s − 3/7·49-s − 1.38·52-s − 4.94·53-s + 3.58·61-s + 5/4·64-s + 2.88·69-s + 6.92·75-s − 3.60·79-s + 14·81-s − 2.57·87-s + 2.08·92-s + 5·100-s − 2.38·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.27926210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.27926210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - T )^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( 1 - 5 T^{2} + 15 T^{4} - 35 T^{6} + 15 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 - 2 p T^{2} + 23 T^{4} + 36 T^{6} + 23 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 p T^{2} + 263 T^{4} - 2724 T^{6} + 263 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 35 T^{2} + 16 T^{3} + 35 p T^{4} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 50 T^{2} + 1063 T^{4} - 16988 T^{6} + 1063 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 2 T + 49 T^{2} - 84 T^{3} + 49 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 10 T^{2} + 17 p T^{4} + 29364 T^{6} + 17 p^{3} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 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 | \( 1 - 114 T^{2} + 4031 T^{4} - 84716 T^{6} + 4031 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 16 T + 161 T^{2} + 1248 T^{3} + 161 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 226 T^{2} + 23391 T^{4} - 1405228 T^{6} + 23391 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 6 T + p T^{2} )^{6} \) |
| 59 | \( 1 - 314 T^{2} + 42855 T^{4} - 3281564 T^{6} + 42855 p^{2} T^{8} - 314 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 14 T + 11 T^{2} + 684 T^{3} + 11 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 274 T^{2} + 36103 T^{4} - 2978332 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 94 T^{2} + 12111 T^{4} - 940276 T^{6} + 12111 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 166 T^{2} + 7231 T^{4} - 21268 T^{6} + 7231 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 16 T + 309 T^{2} + 2608 T^{3} + 309 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 90 T^{2} + 17495 T^{4} - 1240220 T^{6} + 17495 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 322 T^{2} + 50175 T^{4} - 5234188 T^{6} + 50175 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 550 T^{2} + 128975 T^{4} - 16496340 T^{6} + 128975 p^{2} T^{8} - 550 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−6.87077313329531232272053978931, −6.56506880454803847528069253209, −6.25347801892522008576555988528, −5.95999730114951667267006316850, −5.94391882187187924690171256339, −5.88370679230778932494542565513, −5.07896967136810093548979957172, −5.03503493557321978763522096659, −4.98543438906148982988847618257, −4.78811076197096348677920127861, −4.55756201184173832506471217065, −4.34118793921069571939102553210, −3.87614953088857811018814751358, −3.76025364319328962582934486898, −3.41077632709286330860510459863, −3.16320434466550192932125757184, −3.09317382451000324356072312721, −3.00785803725823986099478664975, −2.96676757074889381464756178226, −2.40523797121673712154553993532, −2.14084616359296801334897363508, −1.97851998649247369470744504369, −1.85970556634421356119870076599, −1.52575099620419627149295276262, −1.16562027456661869061557140713,
1.16562027456661869061557140713, 1.52575099620419627149295276262, 1.85970556634421356119870076599, 1.97851998649247369470744504369, 2.14084616359296801334897363508, 2.40523797121673712154553993532, 2.96676757074889381464756178226, 3.00785803725823986099478664975, 3.09317382451000324356072312721, 3.16320434466550192932125757184, 3.41077632709286330860510459863, 3.76025364319328962582934486898, 3.87614953088857811018814751358, 4.34118793921069571939102553210, 4.55756201184173832506471217065, 4.78811076197096348677920127861, 4.98543438906148982988847618257, 5.03503493557321978763522096659, 5.07896967136810093548979957172, 5.88370679230778932494542565513, 5.94391882187187924690171256339, 5.95999730114951667267006316850, 6.25347801892522008576555988528, 6.56506880454803847528069253209, 6.87077313329531232272053978931
Plot not available for L-functions of degree greater than 10.
