| L(s) = 1 | + 0.262·2-s + 0.849·3-s − 1.93·4-s − 1.83·5-s + 0.223·6-s − 0.312·7-s − 1.03·8-s − 2.27·9-s − 0.482·10-s − 11-s − 1.64·12-s − 1.17·13-s − 0.0822·14-s − 1.55·15-s + 3.59·16-s + 2.54·17-s − 0.598·18-s + 7.32·19-s + 3.54·20-s − 0.265·21-s − 0.262·22-s − 0.146·23-s − 0.877·24-s − 1.63·25-s − 0.309·26-s − 4.48·27-s + 0.604·28-s + ⋯ |
| L(s) = 1 | + 0.185·2-s + 0.490·3-s − 0.965·4-s − 0.820·5-s + 0.0911·6-s − 0.118·7-s − 0.365·8-s − 0.759·9-s − 0.152·10-s − 0.301·11-s − 0.473·12-s − 0.326·13-s − 0.0219·14-s − 0.402·15-s + 0.897·16-s + 0.617·17-s − 0.141·18-s + 1.68·19-s + 0.792·20-s − 0.0580·21-s − 0.0560·22-s − 0.0304·23-s − 0.179·24-s − 0.326·25-s − 0.0606·26-s − 0.862·27-s + 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.191136021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.191136021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 0.262T + 2T^{2} \) |
| 3 | \( 1 - 0.849T + 3T^{2} \) |
| 5 | \( 1 + 1.83T + 5T^{2} \) |
| 7 | \( 1 + 0.312T + 7T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 2.54T + 17T^{2} \) |
| 19 | \( 1 - 7.32T + 19T^{2} \) |
| 23 | \( 1 + 0.146T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 - 0.837T + 31T^{2} \) |
| 37 | \( 1 - 4.83T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 8.06T + 43T^{2} \) |
| 47 | \( 1 + 2.40T + 47T^{2} \) |
| 53 | \( 1 - 4.93T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 7.24T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 3.06T + 71T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 4.53T + 83T^{2} \) |
| 89 | \( 1 - 9.16T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−9.448514711860040410357102398902, −8.725944936757597120401407617935, −7.82500545558702783431205034730, −7.57864486006768561923971611619, −6.02737054616026580286563320017, −5.29531837329434952212202062533, −4.35912028758539042010141871577, −3.46221261777812372839191042302, −2.75535318110625574169073790360, −0.74133751435026499691843321691,
0.74133751435026499691843321691, 2.75535318110625574169073790360, 3.46221261777812372839191042302, 4.35912028758539042010141871577, 5.29531837329434952212202062533, 6.02737054616026580286563320017, 7.57864486006768561923971611619, 7.82500545558702783431205034730, 8.725944936757597120401407617935, 9.448514711860040410357102398902
