| L(s) = 1 | − 1.93·2-s + 1.76·4-s − 1.45·5-s − 4.17·7-s + 0.458·8-s + 2.81·10-s + 3.52·11-s − 3.57·13-s + 8.09·14-s − 4.41·16-s − 3.85·17-s + 19-s − 2.56·20-s − 6.84·22-s + 2.10·23-s − 2.88·25-s + 6.94·26-s − 7.36·28-s − 4.78·29-s + 3.51·31-s + 7.65·32-s + 7.47·34-s + 6.06·35-s + 1.52·37-s − 1.93·38-s − 0.666·40-s + 1.32·41-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.881·4-s − 0.649·5-s − 1.57·7-s + 0.162·8-s + 0.891·10-s + 1.06·11-s − 0.992·13-s + 2.16·14-s − 1.10·16-s − 0.934·17-s + 0.229·19-s − 0.573·20-s − 1.45·22-s + 0.439·23-s − 0.577·25-s + 1.36·26-s − 1.39·28-s − 0.888·29-s + 0.631·31-s + 1.35·32-s + 1.28·34-s + 1.02·35-s + 0.250·37-s − 0.314·38-s − 0.105·40-s + 0.206·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 7 | \( 1 + 4.17T + 7T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 + 3.57T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 23 | \( 1 - 2.10T + 23T^{2} \) |
| 29 | \( 1 + 4.78T + 29T^{2} \) |
| 31 | \( 1 - 3.51T + 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 - 1.32T + 41T^{2} \) |
| 43 | \( 1 - 7.20T + 43T^{2} \) |
| 53 | \( 1 - 4.61T + 53T^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 - 5.42T + 61T^{2} \) |
| 67 | \( 1 + 7.86T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41793478385591782610749460269, −7.03718574082190058229230289154, −6.49690660334356287243390560629, −5.61693057598780411375236178322, −4.40284136103199823189526567568, −3.91766798163220544471163718133, −2.92029562445569683847221900870, −2.08351330332679670696218018766, −0.816059929684019000126351134982, 0,
0.816059929684019000126351134982, 2.08351330332679670696218018766, 2.92029562445569683847221900870, 3.91766798163220544471163718133, 4.40284136103199823189526567568, 5.61693057598780411375236178322, 6.49690660334356287243390560629, 7.03718574082190058229230289154, 7.41793478385591782610749460269
