| L(s) = 1 | + 1.48·2-s + 0.216·4-s + 2.91·5-s + 7-s − 2.65·8-s + 4.33·10-s − 4.14·11-s + 3.36·13-s + 1.48·14-s − 4.38·16-s − 4.79·17-s + 0.147·19-s + 0.631·20-s − 6.17·22-s + 5.23·23-s + 3.47·25-s + 5.01·26-s + 0.216·28-s + 1.23·29-s + 6.14·31-s − 1.22·32-s − 7.14·34-s + 2.91·35-s + 7.83·37-s + 0.220·38-s − 7.73·40-s − 0.0490·41-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 0.108·4-s + 1.30·5-s + 0.377·7-s − 0.938·8-s + 1.37·10-s − 1.25·11-s + 0.933·13-s + 0.397·14-s − 1.09·16-s − 1.16·17-s + 0.0339·19-s + 0.141·20-s − 1.31·22-s + 1.09·23-s + 0.695·25-s + 0.983·26-s + 0.0409·28-s + 0.228·29-s + 1.10·31-s − 0.215·32-s − 1.22·34-s + 0.492·35-s + 1.28·37-s + 0.0357·38-s − 1.22·40-s − 0.00766·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.028049084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.028049084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 5 | \( 1 - 2.91T + 5T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 0.147T + 19T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 - 6.14T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 + 0.0490T + 41T^{2} \) |
| 43 | \( 1 + 8.18T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 + 4.90T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 1.83T + 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 - 7.33T + 73T^{2} \) |
| 79 | \( 1 - 6.96T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 0.471T + 97T^{2} \) |
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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
−7.916656351645875397370161080064, −6.62292948183095868729963872981, −6.43190587186325270531226296260, −5.49872945201142467497668289047, −5.13104077364198694826783739931, −4.49661902068026009616723541662, −3.56749936270012538000073012891, −2.63800565215346419372200704552, −2.17149393072623322823613206309, −0.844986161602685709467057718943,
0.844986161602685709467057718943, 2.17149393072623322823613206309, 2.63800565215346419372200704552, 3.56749936270012538000073012891, 4.49661902068026009616723541662, 5.13104077364198694826783739931, 5.49872945201142467497668289047, 6.43190587186325270531226296260, 6.62292948183095868729963872981, 7.916656351645875397370161080064
