| L(s) = 1 | − 2-s − 3-s + 4-s − 3.29·5-s + 6-s + 2.46·7-s − 8-s + 9-s + 3.29·10-s + 5.48·11-s − 12-s − 0.169·13-s − 2.46·14-s + 3.29·15-s + 16-s + 0.362·17-s − 18-s + 4.85·19-s − 3.29·20-s − 2.46·21-s − 5.48·22-s + 23-s + 24-s + 5.85·25-s + 0.169·26-s − 27-s + 2.46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.47·5-s + 0.408·6-s + 0.931·7-s − 0.353·8-s + 0.333·9-s + 1.04·10-s + 1.65·11-s − 0.288·12-s − 0.0470·13-s − 0.658·14-s + 0.850·15-s + 0.250·16-s + 0.0879·17-s − 0.235·18-s + 1.11·19-s − 0.736·20-s − 0.538·21-s − 1.16·22-s + 0.208·23-s + 0.204·24-s + 1.17·25-s + 0.0332·26-s − 0.192·27-s + 0.465·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067834911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067834911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 - 5.48T + 11T^{2} \) |
| 13 | \( 1 + 0.169T + 13T^{2} \) |
| 17 | \( 1 - 0.362T + 17T^{2} \) |
| 19 | \( 1 - 4.85T + 19T^{2} \) |
| 31 | \( 1 + 3.48T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 1.53T + 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 9.27T + 61T^{2} \) |
| 67 | \( 1 + 6.34T + 67T^{2} \) |
| 71 | \( 1 + 6.07T + 71T^{2} \) |
| 73 | \( 1 + 3.86T + 73T^{2} \) |
| 79 | \( 1 + 3.38T + 79T^{2} \) |
| 83 | \( 1 - 5.55T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.354326756415178472931518247419, −7.55586794163289647954787963463, −7.37753125595047426506835233865, −6.37670527755221752939393554408, −5.57847806407679861399926157271, −4.43610033059010684345837261443, −4.06031899064603669674149599849, −2.98834407728878027897381962259, −1.49845518981383636821181098975, −0.75341244899967336848972577795,
0.75341244899967336848972577795, 1.49845518981383636821181098975, 2.98834407728878027897381962259, 4.06031899064603669674149599849, 4.43610033059010684345837261443, 5.57847806407679861399926157271, 6.37670527755221752939393554408, 7.37753125595047426506835233865, 7.55586794163289647954787963463, 8.354326756415178472931518247419
