L(s) = 1 | − 2-s − 0.804·3-s + 4-s − 3.18·5-s + 0.804·6-s − 7-s − 8-s − 2.35·9-s + 3.18·10-s − 6.09·11-s − 0.804·12-s + 4.76·13-s + 14-s + 2.55·15-s + 16-s + 4.44·17-s + 2.35·18-s − 3.99·19-s − 3.18·20-s + 0.804·21-s + 6.09·22-s − 5.05·23-s + 0.804·24-s + 5.13·25-s − 4.76·26-s + 4.30·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.464·3-s + 0.5·4-s − 1.42·5-s + 0.328·6-s − 0.377·7-s − 0.353·8-s − 0.784·9-s + 1.00·10-s − 1.83·11-s − 0.232·12-s + 1.32·13-s + 0.267·14-s + 0.660·15-s + 0.250·16-s + 1.07·17-s + 0.554·18-s − 0.915·19-s − 0.711·20-s + 0.175·21-s + 1.29·22-s − 1.05·23-s + 0.164·24-s + 1.02·25-s − 0.935·26-s + 0.828·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 + 0.804T + 3T^{2} \) |
| 5 | \( 1 + 3.18T + 5T^{2} \) |
| 11 | \( 1 + 6.09T + 11T^{2} \) |
| 13 | \( 1 - 4.76T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 3.99T + 19T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 - 0.204T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 5.54T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 0.201T + 53T^{2} \) |
| 59 | \( 1 - 7.13T + 59T^{2} \) |
| 61 | \( 1 - 7.83T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 4.06T + 79T^{2} \) |
| 83 | \( 1 + 7.35T + 83T^{2} \) |
| 89 | \( 1 - 1.32T + 89T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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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
−7.972640713185883852261686272546, −7.26408870785679960043961403920, −6.20798620342098498674768014978, −5.82710075653279335602809226229, −4.88371414536842366380713004183, −3.90045770041498420918377945885, −3.18817003168099956963712098937, −2.39447295996792283886839873248, −0.809788001044415387845941703292, 0,
0.809788001044415387845941703292, 2.39447295996792283886839873248, 3.18817003168099956963712098937, 3.90045770041498420918377945885, 4.88371414536842366380713004183, 5.82710075653279335602809226229, 6.20798620342098498674768014978, 7.26408870785679960043961403920, 7.972640713185883852261686272546