| L(s) = 1 | + 2-s − 1.05·3-s + 4-s − 2.35·5-s − 1.05·6-s − 2.98·7-s + 8-s − 1.87·9-s − 2.35·10-s − 3.05·11-s − 1.05·12-s − 7.10·13-s − 2.98·14-s + 2.49·15-s + 16-s − 2.73·17-s − 1.87·18-s + 0.683·19-s − 2.35·20-s + 3.16·21-s − 3.05·22-s − 7.43·23-s − 1.05·24-s + 0.538·25-s − 7.10·26-s + 5.16·27-s − 2.98·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.611·3-s + 0.5·4-s − 1.05·5-s − 0.432·6-s − 1.12·7-s + 0.353·8-s − 0.626·9-s − 0.744·10-s − 0.921·11-s − 0.305·12-s − 1.97·13-s − 0.798·14-s + 0.643·15-s + 0.250·16-s − 0.663·17-s − 0.443·18-s + 0.156·19-s − 0.526·20-s + 0.690·21-s − 0.651·22-s − 1.55·23-s − 0.216·24-s + 0.107·25-s − 1.39·26-s + 0.993·27-s − 0.564·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3202939956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3202939956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 1.05T + 3T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 7.10T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 - 0.683T + 19T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 + 2.48T + 29T^{2} \) |
| 31 | \( 1 - 8.43T + 31T^{2} \) |
| 37 | \( 1 - 4.99T + 37T^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 - 9.62T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 7.86T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 - 4.88T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 5.37T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 6.11T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 + 6.00T + 97T^{2} \) |
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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
−8.036058664532928505232537988226, −7.72795819392194003772236595180, −6.83586283802471476446435162251, −6.16648430774389867362639929019, −5.44894494461508515349349222435, −4.63629611886566498890357374408, −4.02288330200152460765119855631, −2.87493589745479286201933280899, −2.48142626007485249475053972580, −0.27204914028088355716660822396,
0.27204914028088355716660822396, 2.48142626007485249475053972580, 2.87493589745479286201933280899, 4.02288330200152460765119855631, 4.63629611886566498890357374408, 5.44894494461508515349349222435, 6.16648430774389867362639929019, 6.83586283802471476446435162251, 7.72795819392194003772236595180, 8.036058664532928505232537988226
