| L(s) = 1 | + 3·4-s − 4·5-s − 9-s + 2·11-s + 5·16-s − 12·20-s + 11·25-s − 12·31-s − 3·36-s − 12·41-s + 6·44-s + 4·45-s − 49-s − 8·55-s + 4·61-s + 3·64-s + 16·71-s − 4·79-s − 20·80-s + 81-s + 24·89-s − 2·99-s + 33·100-s − 28·101-s + 36·109-s + 3·121-s − 36·124-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1.78·5-s − 1/3·9-s + 0.603·11-s + 5/4·16-s − 2.68·20-s + 11/5·25-s − 2.15·31-s − 1/2·36-s − 1.87·41-s + 0.904·44-s + 0.596·45-s − 1/7·49-s − 1.07·55-s + 0.512·61-s + 3/8·64-s + 1.89·71-s − 0.450·79-s − 2.23·80-s + 1/9·81-s + 2.54·89-s − 0.201·99-s + 3.29·100-s − 2.78·101-s + 3.44·109-s + 3/11·121-s − 3.23·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.689591229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.689591229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.19919550171989171630485507957, −9.535734241313627084831169393116, −9.110530735907319921264441673417, −8.626337533542103104125760790691, −8.344588168152005753671927811958, −7.69727475863075152313665627912, −7.65130469186450579823583422549, −6.97251080753238389549594678385, −6.91541796721221392793059397525, −6.38533601026859193980550670572, −5.93670099945082156465073246983, −5.11299480821682757260827718333, −5.07551724314611903166488882426, −4.06802003649711077096787991795, −3.82994388585028736148059895500, −3.30484578069762892950136960942, −2.94648901688628314398379962575, −2.10154163342333559636944188902, −1.61859760466338375287059253568, −0.55170394821371558821969932425,
0.55170394821371558821969932425, 1.61859760466338375287059253568, 2.10154163342333559636944188902, 2.94648901688628314398379962575, 3.30484578069762892950136960942, 3.82994388585028736148059895500, 4.06802003649711077096787991795, 5.07551724314611903166488882426, 5.11299480821682757260827718333, 5.93670099945082156465073246983, 6.38533601026859193980550670572, 6.91541796721221392793059397525, 6.97251080753238389549594678385, 7.65130469186450579823583422549, 7.69727475863075152313665627912, 8.344588168152005753671927811958, 8.626337533542103104125760790691, 9.110530735907319921264441673417, 9.535734241313627084831169393116, 10.19919550171989171630485507957
