| L(s) = 1 | + 0.0898·2-s − 1.99·4-s + 4.36·7-s − 0.358·8-s − 4.39·11-s − 1.98·13-s + 0.391·14-s + 3.95·16-s − 0.997·17-s + 1.35·19-s − 0.394·22-s − 2.35·23-s − 0.177·26-s − 8.68·28-s + 7.97·29-s − 3.67·31-s + 1.07·32-s − 0.0895·34-s − 1.43·37-s + 0.121·38-s + 5.98·41-s + 2.68·43-s + 8.74·44-s − 0.211·46-s − 10.9·47-s + 12.0·49-s + 3.94·52-s + ⋯ |
| L(s) = 1 | + 0.0635·2-s − 0.995·4-s + 1.64·7-s − 0.126·8-s − 1.32·11-s − 0.549·13-s + 0.104·14-s + 0.987·16-s − 0.241·17-s + 0.309·19-s − 0.0840·22-s − 0.491·23-s − 0.0349·26-s − 1.64·28-s + 1.48·29-s − 0.660·31-s + 0.189·32-s − 0.0153·34-s − 0.236·37-s + 0.0196·38-s + 0.934·41-s + 0.409·43-s + 1.31·44-s − 0.0312·46-s − 1.59·47-s + 1.71·49-s + 0.547·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.0898T + 2T^{2} \) |
| 7 | \( 1 - 4.36T + 7T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 13 | \( 1 + 1.98T + 13T^{2} \) |
| 17 | \( 1 + 0.997T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 + 3.67T + 31T^{2} \) |
| 37 | \( 1 + 1.43T + 37T^{2} \) |
| 41 | \( 1 - 5.98T + 41T^{2} \) |
| 43 | \( 1 - 2.68T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 6.68T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 7.32T + 71T^{2} \) |
| 73 | \( 1 + 0.424T + 73T^{2} \) |
| 79 | \( 1 + 6.35T + 79T^{2} \) |
| 83 | \( 1 + 0.737T + 83T^{2} \) |
| 89 | \( 1 - 9.78T + 89T^{2} \) |
| 97 | \( 1 - 0.0337T + 97T^{2} \) |
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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
−7.918053509092981058550553567827, −7.41675319454720887506624953196, −6.17412746490964607465187509651, −5.33457275760517876800337513548, −4.80288454386320904064426318599, −4.45639044384356649940342883373, −3.27476741201105000318364111736, −2.32151689946617855012932768954, −1.29710652646225860579383380477, 0,
1.29710652646225860579383380477, 2.32151689946617855012932768954, 3.27476741201105000318364111736, 4.45639044384356649940342883373, 4.80288454386320904064426318599, 5.33457275760517876800337513548, 6.17412746490964607465187509651, 7.41675319454720887506624953196, 7.918053509092981058550553567827
