| L(s) = 1 | − 0.414·2-s − 0.414·3-s − 1.82·4-s + 2·5-s + 0.171·6-s + 1.82·7-s + 1.58·8-s − 2.82·9-s − 0.828·10-s + 0.757·12-s + 5·13-s − 0.757·14-s − 0.828·15-s + 3·16-s + 1.58·17-s + 1.17·18-s + 3.24·19-s − 3.65·20-s − 0.757·21-s + 8.82·23-s − 0.656·24-s − 25-s − 2.07·26-s + 2.41·27-s − 3.34·28-s + 29-s + 0.343·30-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.239·3-s − 0.914·4-s + 0.894·5-s + 0.0700·6-s + 0.691·7-s + 0.560·8-s − 0.942·9-s − 0.261·10-s + 0.218·12-s + 1.38·13-s − 0.202·14-s − 0.213·15-s + 0.750·16-s + 0.384·17-s + 0.276·18-s + 0.743·19-s − 0.817·20-s − 0.165·21-s + 1.84·23-s − 0.134·24-s − 0.200·25-s − 0.406·26-s + 0.464·27-s − 0.631·28-s + 0.185·29-s + 0.0626·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.654615961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.654615961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 31 | \( 1 + 0.828T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 0.414T + 43T^{2} \) |
| 47 | \( 1 + 0.828T + 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 + 0.343T + 59T^{2} \) |
| 61 | \( 1 + 8.89T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 7T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 0.757T + 97T^{2} \) |
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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
−8.445361455226514549143857845182, −8.044817370407442611537239136355, −6.95647873757739248726279849948, −6.06964454767894990182097768494, −5.29840142736284275786883936978, −5.05362436177255829529506537724, −3.80533769671276014998152164596, −3.01930448942650949189552672391, −1.66410408567194486387988019198, −0.851703007429084841432176246676,
0.851703007429084841432176246676, 1.66410408567194486387988019198, 3.01930448942650949189552672391, 3.80533769671276014998152164596, 5.05362436177255829529506537724, 5.29840142736284275786883936978, 6.06964454767894990182097768494, 6.95647873757739248726279849948, 8.044817370407442611537239136355, 8.445361455226514549143857845182
