| L(s) = 1 | − 2-s + 4-s + 1.15i·5-s − 7-s − 8-s − 1.15i·10-s − 2.90i·11-s − 1.38i·13-s + 14-s + 16-s − 2.90i·17-s + (3.79 − 2.14i)19-s + 1.15i·20-s + 2.90i·22-s + 7.14i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.516i·5-s − 0.377·7-s − 0.353·8-s − 0.365i·10-s − 0.876i·11-s − 0.383i·13-s + 0.267·14-s + 0.250·16-s − 0.704i·17-s + (0.870 − 0.492i)19-s + 0.258i·20-s + 0.619i·22-s + 1.49i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7596294993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7596294993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + (-3.79 + 2.14i)T \) |
| good | 5 | \( 1 - 1.15iT - 5T^{2} \) |
| 11 | \( 1 + 2.90iT - 11T^{2} \) |
| 13 | \( 1 + 1.38iT - 13T^{2} \) |
| 17 | \( 1 + 2.90iT - 17T^{2} \) |
| 23 | \( 1 - 7.14iT - 23T^{2} \) |
| 29 | \( 1 + 4.66T + 29T^{2} \) |
| 31 | \( 1 - 4.06iT - 31T^{2} \) |
| 37 | \( 1 + 6.55iT - 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 2.04T + 43T^{2} \) |
| 47 | \( 1 + 2.76iT - 47T^{2} \) |
| 53 | \( 1 + 7.39T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.29T + 61T^{2} \) |
| 67 | \( 1 + 9.91iT - 67T^{2} \) |
| 71 | \( 1 + 6.06T + 71T^{2} \) |
| 73 | \( 1 - 9.37T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 14.5iT - 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + 3.92iT - 97T^{2} \) |
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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
−8.991284510765432676353832254610, −7.911142995724033212553115306251, −7.31918116266606818118979349300, −6.63143817567202720842767614865, −5.72992986208555849365122175993, −4.99846117907972895752553536937, −3.31112862158275651186014667865, −3.17246725199094981257075067431, −1.70322534066303850513589124078, −0.34897871331763152470663298116,
1.19846467469450061405261151503, 2.20330055326520204609258480559, 3.35893331756626149378883933572, 4.41029133553639128278686465259, 5.23646956155696811884194496129, 6.31293057590650373539713482598, 6.87286643968984711023345597778, 7.79834388315267894528592264477, 8.439034754138502793175270401236, 9.186931152604920604564400468320
