| L(s) = 1 | − 2.51·2-s − 2.35·3-s + 4.30·4-s + 5-s + 5.90·6-s + 7-s − 5.77·8-s + 2.53·9-s − 2.51·10-s − 0.675·11-s − 10.1·12-s + 4.04·13-s − 2.51·14-s − 2.35·15-s + 5.89·16-s − 5.02·17-s − 6.35·18-s − 6.34·19-s + 4.30·20-s − 2.35·21-s + 1.69·22-s + 6.41·23-s + 13.5·24-s + 25-s − 10.1·26-s + 1.10·27-s + 4.30·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 1.35·3-s + 2.15·4-s + 0.447·5-s + 2.40·6-s + 0.377·7-s − 2.04·8-s + 0.843·9-s − 0.793·10-s − 0.203·11-s − 2.91·12-s + 1.12·13-s − 0.670·14-s − 0.607·15-s + 1.47·16-s − 1.21·17-s − 1.49·18-s − 1.45·19-s + 0.961·20-s − 0.513·21-s + 0.361·22-s + 1.33·23-s + 2.77·24-s + 0.200·25-s − 1.98·26-s + 0.212·27-s + 0.812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 + 2.35T + 3T^{2} \) |
| 11 | \( 1 + 0.675T + 11T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 - 6.41T + 23T^{2} \) |
| 29 | \( 1 + 5.00T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 - 5.15T + 43T^{2} \) |
| 47 | \( 1 + 1.84T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 0.969T + 59T^{2} \) |
| 61 | \( 1 + 8.30T + 61T^{2} \) |
| 67 | \( 1 + 8.10T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 - 0.219T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 1.27T + 89T^{2} \) |
| 97 | \( 1 - 0.0392T + 97T^{2} \) |
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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
−7.48003498094471120335811404713, −6.79154029533858961547774591149, −6.34337643991683033890535216703, −5.72684236615822888097196667987, −4.91714303226950948930991745381, −3.96359117747037698008413163714, −2.56945857130897499215561091550, −1.79457182173202981129578395024, −0.942184630248065029584313365641, 0,
0.942184630248065029584313365641, 1.79457182173202981129578395024, 2.56945857130897499215561091550, 3.96359117747037698008413163714, 4.91714303226950948930991745381, 5.72684236615822888097196667987, 6.34337643991683033890535216703, 6.79154029533858961547774591149, 7.48003498094471120335811404713
