| L(s) = 1 | + 2.51·2-s + 4.31·4-s − 1.65·5-s + 0.864·7-s + 5.82·8-s − 4.14·10-s − 3.73·11-s − 5.68·13-s + 2.17·14-s + 5.99·16-s − 5.96·17-s + 1.24·19-s − 7.12·20-s − 9.38·22-s − 7.12·23-s − 2.27·25-s − 14.2·26-s + 3.73·28-s + 5.91·29-s + 5.20·31-s + 3.42·32-s − 14.9·34-s − 1.42·35-s − 10.0·37-s + 3.13·38-s − 9.61·40-s − 0.350·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.15·4-s − 0.738·5-s + 0.326·7-s + 2.05·8-s − 1.31·10-s − 1.12·11-s − 1.57·13-s + 0.580·14-s + 1.49·16-s − 1.44·17-s + 0.285·19-s − 1.59·20-s − 2.00·22-s − 1.48·23-s − 0.454·25-s − 2.79·26-s + 0.705·28-s + 1.09·29-s + 0.934·31-s + 0.605·32-s − 2.56·34-s − 0.241·35-s − 1.66·37-s + 0.507·38-s − 1.51·40-s − 0.0546·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 - 0.864T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 - 5.91T + 29T^{2} \) |
| 31 | \( 1 - 5.20T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 0.350T + 41T^{2} \) |
| 43 | \( 1 - 9.82T + 43T^{2} \) |
| 47 | \( 1 + 0.398T + 47T^{2} \) |
| 53 | \( 1 - 4.07T + 53T^{2} \) |
| 59 | \( 1 + 9.96T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 - 5.51T + 67T^{2} \) |
| 71 | \( 1 - 5.20T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79769224535254203260656859802, −7.19352274271136374087716574147, −6.47785753554433960335979141419, −5.59589386036580148157336180113, −4.81796126336858689679345105767, −4.48822321139401667057545058828, −3.61653367652601060346039428036, −2.59168888831911085576383781295, −2.13967344981942956554246299421, 0,
2.13967344981942956554246299421, 2.59168888831911085576383781295, 3.61653367652601060346039428036, 4.48822321139401667057545058828, 4.81796126336858689679345105767, 5.59589386036580148157336180113, 6.47785753554433960335979141419, 7.19352274271136374087716574147, 7.79769224535254203260656859802
