| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 5·5-s − 6·6-s − 4·7-s + 8·8-s + 9·9-s + 10·10-s − 12·11-s − 12·12-s − 46·13-s − 8·14-s − 15·15-s + 16·16-s − 102·17-s + 18·18-s + 19·19-s + 20·20-s + 12·21-s − 24·22-s − 84·23-s − 24·24-s + 25·25-s − 92·26-s − 27·27-s − 16·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.215·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.328·11-s − 0.288·12-s − 0.981·13-s − 0.152·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.124·21-s − 0.232·22-s − 0.761·23-s − 0.204·24-s + 1/5·25-s − 0.693·26-s − 0.192·27-s − 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 46 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 222 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 160 T + p^{3} T^{2} \) |
| 47 | \( 1 - 36 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 516 T + p^{3} T^{2} \) |
| 61 | \( 1 + 346 T + p^{3} T^{2} \) |
| 67 | \( 1 + 700 T + p^{3} T^{2} \) |
| 71 | \( 1 + 480 T + p^{3} T^{2} \) |
| 73 | \( 1 - 338 T + p^{3} T^{2} \) |
| 79 | \( 1 - 248 T + p^{3} T^{2} \) |
| 83 | \( 1 - 720 T + p^{3} T^{2} \) |
| 89 | \( 1 + 30 T + p^{3} T^{2} \) |
| 97 | \( 1 - 614 T + p^{3} T^{2} \) |
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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
−10.15161489645659318251131537976, −9.164554686971865864696106891946, −7.928114546509751992338996896094, −6.82558115730748757558665770313, −6.23000981533018163497920370328, −5.10619318011660134023242757624, −4.46892795526603772070593073613, −2.99169670288100153388022049135, −1.85030126629743760662069071364, 0,
1.85030126629743760662069071364, 2.99169670288100153388022049135, 4.46892795526603772070593073613, 5.10619318011660134023242757624, 6.23000981533018163497920370328, 6.82558115730748757558665770313, 7.928114546509751992338996896094, 9.164554686971865864696106891946, 10.15161489645659318251131537976
