| L(s) = 1 | − 0.358·2-s + 3-s − 1.87·4-s − 5-s − 0.358·6-s + 2.43·7-s + 1.38·8-s + 9-s + 0.358·10-s + 4.36·11-s − 1.87·12-s − 5.75·13-s − 0.873·14-s − 15-s + 3.24·16-s + 0.243·17-s − 0.358·18-s + 3.07·19-s + 1.87·20-s + 2.43·21-s − 1.56·22-s + 1.38·24-s + 25-s + 2.06·26-s + 27-s − 4.55·28-s + 2.57·29-s + ⋯ |
| L(s) = 1 | − 0.253·2-s + 0.577·3-s − 0.935·4-s − 0.447·5-s − 0.146·6-s + 0.920·7-s + 0.490·8-s + 0.333·9-s + 0.113·10-s + 1.31·11-s − 0.540·12-s − 1.59·13-s − 0.233·14-s − 0.258·15-s + 0.811·16-s + 0.0590·17-s − 0.0844·18-s + 0.705·19-s + 0.418·20-s + 0.531·21-s − 0.333·22-s + 0.283·24-s + 0.200·25-s + 0.404·26-s + 0.192·27-s − 0.861·28-s + 0.477·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.907581604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.907581604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.358T + 2T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 + 5.75T + 13T^{2} \) |
| 17 | \( 1 - 0.243T + 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 29 | \( 1 - 2.57T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 + 4.66T + 37T^{2} \) |
| 41 | \( 1 - 4.81T + 41T^{2} \) |
| 43 | \( 1 - 9.78T + 43T^{2} \) |
| 47 | \( 1 + 5.51T + 47T^{2} \) |
| 53 | \( 1 - 4.31T + 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 - 1.67T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 6.99T + 73T^{2} \) |
| 79 | \( 1 + 6.05T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.87313192754562146444793950983, −7.42846851563798575139712554912, −6.70361159128855585890686497202, −5.55829709219314120739852957244, −4.79451296719967218852351934761, −4.37072107678881582119989141239, −3.64789035216894443340240376987, −2.69645197733347823929784709970, −1.62312392477281631920379026678, −0.74279090704461289365532370771,
0.74279090704461289365532370771, 1.62312392477281631920379026678, 2.69645197733347823929784709970, 3.64789035216894443340240376987, 4.37072107678881582119989141239, 4.79451296719967218852351934761, 5.55829709219314120739852957244, 6.70361159128855585890686497202, 7.42846851563798575139712554912, 7.87313192754562146444793950983
