| L(s) = 1 | + 4.09e3·2-s − 9.79e4·3-s + 1.67e7·4-s − 4.01e8·6-s + 4.08e10·7-s + 6.87e10·8-s − 8.37e11·9-s − 1.45e13·11-s − 1.64e12·12-s − 8.78e13·13-s + 1.67e14·14-s + 2.81e14·16-s + 2.65e15·17-s − 3.43e15·18-s − 1.39e16·19-s − 4.00e15·21-s − 5.94e16·22-s − 8.58e16·23-s − 6.73e15·24-s − 3.59e17·26-s + 1.65e17·27-s + 6.85e17·28-s + 2.08e18·29-s + 2.66e18·31-s + 1.15e18·32-s + 1.42e18·33-s + 1.08e19·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.106·3-s + 1/2·4-s − 0.0752·6-s + 1.11·7-s + 0.353·8-s − 0.988·9-s − 1.39·11-s − 0.0532·12-s − 1.04·13-s + 0.789·14-s + 1/4·16-s + 1.10·17-s − 0.699·18-s − 1.45·19-s − 0.118·21-s − 0.985·22-s − 0.816·23-s − 0.0376·24-s − 0.739·26-s + 0.211·27-s + 0.558·28-s + 1.09·29-s + 0.607·31-s + 0.176·32-s + 0.148·33-s + 0.781·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(2.916552461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.916552461\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{12} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3628 p^{3} T + p^{25} T^{2} \) |
| 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
−11.02656251924402253250799584478, −10.08267501699160350710920492597, −8.278428694203786333297525736239, −7.70418321935235883830277414275, −6.10319740238569068084629756040, −5.18994914148217405128095737052, −4.40608122956210948873992596926, −2.80866166479246337361156395048, −2.16729056947531441903463375907, −0.61892030395535338017016782183,
0.61892030395535338017016782183, 2.16729056947531441903463375907, 2.80866166479246337361156395048, 4.40608122956210948873992596926, 5.18994914148217405128095737052, 6.10319740238569068084629756040, 7.70418321935235883830277414275, 8.278428694203786333297525736239, 10.08267501699160350710920492597, 11.02656251924402253250799584478
