| L(s) = 1 | − 9.79e4·3-s + 3.41e8·5-s + 4.08e10·7-s − 8.37e11·9-s + 1.45e13·11-s + 8.78e13·13-s − 3.34e13·15-s − 2.65e15·17-s + 1.39e16·19-s − 4.00e15·21-s − 8.58e16·23-s − 1.81e17·25-s + 1.65e17·27-s + 2.08e18·29-s − 2.66e18·31-s − 1.42e18·33-s + 1.39e19·35-s − 5.13e19·37-s − 8.60e18·39-s + 2.33e20·41-s + 4.01e19·43-s − 2.85e20·45-s − 2.79e20·47-s + 3.30e20·49-s + 2.60e20·51-s + 4.25e20·53-s + 4.94e21·55-s + ⋯ |
| L(s) = 1 | − 0.106·3-s + 0.624·5-s + 1.11·7-s − 0.988·9-s + 1.39·11-s + 1.04·13-s − 0.0664·15-s − 1.10·17-s + 1.45·19-s − 0.118·21-s − 0.816·23-s − 0.609·25-s + 0.211·27-s + 1.09·29-s − 0.607·31-s − 0.148·33-s + 0.697·35-s − 1.28·37-s − 0.111·39-s + 1.61·41-s + 0.153·43-s − 0.617·45-s − 0.351·47-s + 0.246·49-s + 0.117·51-s + 0.118·53-s + 0.870·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(2.878329400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.878329400\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 3628 p^{3} T + p^{25} T^{2} \) |
| 5 | \( 1 - 13640214 p^{2} T + p^{25} T^{2} \) |
| 7 | \( 1 - 40882637368 T + p^{25} T^{2} \) |
| 11 | \( 1 - 119886135348 p^{2} T + p^{25} T^{2} \) |
| 13 | \( 1 - 87843989537006 T + p^{25} T^{2} \) |
| 17 | \( 1 + 156201521699214 p T + p^{25} T^{2} \) |
| 19 | \( 1 - 736811826531460 p T + p^{25} T^{2} \) |
| 23 | \( 1 + 3732729596697192 p T + p^{25} T^{2} \) |
| 29 | \( 1 - 2080230429601526910 T + p^{25} T^{2} \) |
| 31 | \( 1 + 2663532371302675232 T + p^{25} T^{2} \) |
| 37 | \( 1 + 51379607980315436218 T + p^{25} T^{2} \) |
| 41 | \( 1 - \)\(23\!\cdots\!22\)\( T + p^{25} T^{2} \) |
| 43 | \( 1 - 40133597094729613684 T + p^{25} T^{2} \) |
| 47 | \( 1 + \)\(27\!\cdots\!72\)\( T + p^{25} T^{2} \) |
| 53 | \( 1 - \)\(42\!\cdots\!26\)\( T + p^{25} T^{2} \) |
| 59 | \( 1 - \)\(83\!\cdots\!80\)\( T + p^{25} T^{2} \) |
| 61 | \( 1 - \)\(24\!\cdots\!62\)\( T + p^{25} T^{2} \) |
| 67 | \( 1 - \)\(12\!\cdots\!28\)\( T + p^{25} T^{2} \) |
| 71 | \( 1 - \)\(93\!\cdots\!88\)\( T + p^{25} T^{2} \) |
| 73 | \( 1 - \)\(40\!\cdots\!86\)\( T + p^{25} T^{2} \) |
| 79 | \( 1 - \)\(80\!\cdots\!80\)\( T + p^{25} T^{2} \) |
| 83 | \( 1 + \)\(89\!\cdots\!76\)\( T + p^{25} T^{2} \) |
| 89 | \( 1 - \)\(35\!\cdots\!90\)\( T + p^{25} T^{2} \) |
| 97 | \( 1 + \)\(86\!\cdots\!18\)\( T + p^{25} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.80151443094851795967417640563, −11.84165622855376208152891093725, −11.05326067697398938891461185431, −9.315139448915723247977548701247, −8.257520937008417841471232051112, −6.48098376477412601015058869585, −5.34533215264212542012033861633, −3.81468890268041168450712402979, −2.07720830743184260377037426933, −0.972156471475575857234318561735,
0.972156471475575857234318561735, 2.07720830743184260377037426933, 3.81468890268041168450712402979, 5.34533215264212542012033861633, 6.48098376477412601015058869585, 8.257520937008417841471232051112, 9.315139448915723247977548701247, 11.05326067697398938891461185431, 11.84165622855376208152891093725, 13.80151443094851795967417640563
