| L(s) = 1 | + 2-s + 4-s + 3.43·5-s − 1.90·7-s + 8-s + 3.43·10-s − 5.05·11-s − 4.50·13-s − 1.90·14-s + 16-s + 3.54·17-s − 4.96·19-s + 3.43·20-s − 5.05·22-s + 7.75·23-s + 6.79·25-s − 4.50·26-s − 1.90·28-s − 8.52·29-s − 8.93·31-s + 32-s + 3.54·34-s − 6.53·35-s + 2.08·37-s − 4.96·38-s + 3.43·40-s + 5.04·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.53·5-s − 0.719·7-s + 0.353·8-s + 1.08·10-s − 1.52·11-s − 1.25·13-s − 0.508·14-s + 0.250·16-s + 0.858·17-s − 1.13·19-s + 0.767·20-s − 1.07·22-s + 1.61·23-s + 1.35·25-s − 0.884·26-s − 0.359·28-s − 1.58·29-s − 1.60·31-s + 0.176·32-s + 0.607·34-s − 1.10·35-s + 0.342·37-s − 0.805·38-s + 0.542·40-s + 0.788·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 3.43T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 + 5.05T + 11T^{2} \) |
| 13 | \( 1 + 4.50T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 23 | \( 1 - 7.75T + 23T^{2} \) |
| 29 | \( 1 + 8.52T + 29T^{2} \) |
| 31 | \( 1 + 8.93T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 - 5.04T + 41T^{2} \) |
| 43 | \( 1 + 4.60T + 43T^{2} \) |
| 47 | \( 1 - 5.86T + 47T^{2} \) |
| 53 | \( 1 + 2.76T + 53T^{2} \) |
| 59 | \( 1 + 5.93T + 59T^{2} \) |
| 61 | \( 1 - 5.94T + 61T^{2} \) |
| 67 | \( 1 + 6.16T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 6.74T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−7.32853499833813978743927610034, −6.68353074192795831102260537758, −5.91038981362113331563359742567, −5.27792473720951774942736443860, −5.08879169293720372040583105260, −3.87013884207136466686939833129, −2.84090057679338410976977057251, −2.49254830169972366917743053226, −1.61766363383493931085593227513, 0,
1.61766363383493931085593227513, 2.49254830169972366917743053226, 2.84090057679338410976977057251, 3.87013884207136466686939833129, 5.08879169293720372040583105260, 5.27792473720951774942736443860, 5.91038981362113331563359742567, 6.68353074192795831102260537758, 7.32853499833813978743927610034
