| L(s) = 1 | + 2·5-s + 9-s + 8·11-s + 5·25-s + 24·29-s + 8·31-s + 40·41-s + 2·45-s + 16·55-s − 8·59-s + 4·61-s − 24·79-s + 20·89-s + 8·99-s − 4·101-s − 4·109-s + 38·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + 16·155-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s + 2.41·11-s + 25-s + 4.45·29-s + 1.43·31-s + 6.24·41-s + 0.298·45-s + 2.15·55-s − 1.04·59-s + 0.512·61-s − 2.70·79-s + 2.11·89-s + 0.804·99-s − 0.398·101-s − 0.383·109-s + 3.45·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.14651166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.14651166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 78 T^{2} + 3875 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 82 T^{2} + 1395 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.17691336383326233273012993272, −5.92737364257304510730377100793, −5.82845157108284219333384889279, −5.82680832856455632703707059416, −5.62089721951656408656451595976, −4.99568202989921654313559139755, −4.95794196273389300800434867635, −4.58485734453172520474782885617, −4.54356814214105096915637718030, −4.34758824835669358294309739775, −4.20228014373754662533258764276, −4.04566931337420137652026539519, −3.90947180048060406283874341320, −3.16339255585273611147973998494, −3.05618439753852648063993467595, −3.01522320088461653802407578434, −2.91588936188680976752005186297, −2.51240199560116763114368581130, −2.11794387766076669046191034499, −1.94087666560935250387361907651, −1.77386860575861602152715686459, −1.07331242778245243522132327064, −0.956300214662146171896165375571, −0.845782961258970019901223728095, −0.799101829632764161884040001244,
0.799101829632764161884040001244, 0.845782961258970019901223728095, 0.956300214662146171896165375571, 1.07331242778245243522132327064, 1.77386860575861602152715686459, 1.94087666560935250387361907651, 2.11794387766076669046191034499, 2.51240199560116763114368581130, 2.91588936188680976752005186297, 3.01522320088461653802407578434, 3.05618439753852648063993467595, 3.16339255585273611147973998494, 3.90947180048060406283874341320, 4.04566931337420137652026539519, 4.20228014373754662533258764276, 4.34758824835669358294309739775, 4.54356814214105096915637718030, 4.58485734453172520474782885617, 4.95794196273389300800434867635, 4.99568202989921654313559139755, 5.62089721951656408656451595976, 5.82680832856455632703707059416, 5.82845157108284219333384889279, 5.92737364257304510730377100793, 6.17691336383326233273012993272
