| L(s) = 1 | + 2·2-s − 7·3-s + 4·4-s − 14·6-s + 7·7-s + 8·8-s + 22·9-s − 37·11-s − 28·12-s + 51·13-s + 14·14-s + 16·16-s + 41·17-s + 44·18-s − 108·19-s − 49·21-s − 74·22-s − 70·23-s − 56·24-s + 102·26-s + 35·27-s + 28·28-s − 249·29-s − 134·31-s + 32·32-s + 259·33-s + 82·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·3-s + 1/2·4-s − 0.952·6-s + 0.377·7-s + 0.353·8-s + 0.814·9-s − 1.01·11-s − 0.673·12-s + 1.08·13-s + 0.267·14-s + 1/4·16-s + 0.584·17-s + 0.576·18-s − 1.30·19-s − 0.509·21-s − 0.717·22-s − 0.634·23-s − 0.476·24-s + 0.769·26-s + 0.249·27-s + 0.188·28-s − 1.59·29-s − 0.776·31-s + 0.176·32-s + 1.36·33-s + 0.413·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 37 T + p^{3} T^{2} \) |
| 13 | \( 1 - 51 T + p^{3} T^{2} \) |
| 17 | \( 1 - 41 T + p^{3} T^{2} \) |
| 19 | \( 1 + 108 T + p^{3} T^{2} \) |
| 23 | \( 1 + 70 T + p^{3} T^{2} \) |
| 29 | \( 1 + 249 T + p^{3} T^{2} \) |
| 31 | \( 1 + 134 T + p^{3} T^{2} \) |
| 37 | \( 1 + 334 T + p^{3} T^{2} \) |
| 41 | \( 1 - 206 T + p^{3} T^{2} \) |
| 43 | \( 1 + 376 T + p^{3} T^{2} \) |
| 47 | \( 1 + 287 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 T + p^{3} T^{2} \) |
| 59 | \( 1 + 2 T + p^{3} T^{2} \) |
| 61 | \( 1 + 940 T + p^{3} T^{2} \) |
| 67 | \( 1 - 106 T + p^{3} T^{2} \) |
| 71 | \( 1 - 456 T + p^{3} T^{2} \) |
| 73 | \( 1 - 650 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1239 T + p^{3} T^{2} \) |
| 83 | \( 1 - 428 T + p^{3} T^{2} \) |
| 89 | \( 1 + 220 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1055 T + p^{3} T^{2} \) |
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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
−10.91201038706657979384563392446, −10.18432314911464681880998034574, −8.582266287287243241122366445557, −7.50407217996234788045526692943, −6.31845114566246687509611308594, −5.64127904548606956652399434358, −4.82545181108521761552790457876, −3.58510930211780134307611807312, −1.78803303415401535977087150005, 0,
1.78803303415401535977087150005, 3.58510930211780134307611807312, 4.82545181108521761552790457876, 5.64127904548606956652399434358, 6.31845114566246687509611308594, 7.50407217996234788045526692943, 8.582266287287243241122366445557, 10.18432314911464681880998034574, 10.91201038706657979384563392446
