| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s − 2·9-s + 3·11-s − 12-s − 2·13-s + 14-s + 16-s − 3·17-s + 2·18-s − 7·19-s + 21-s − 3·22-s + 24-s + 2·26-s + 5·27-s − 28-s − 6·29-s − 4·31-s − 32-s − 3·33-s + 3·34-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.904·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.60·19-s + 0.218·21-s − 0.639·22-s + 0.204·24-s + 0.392·26-s + 0.962·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.97618644248142866152725079598, −10.18616620126561124117671237886, −9.056880838216350511563756479245, −8.489865387400252022087843283105, −7.02864082573576697328686349834, −6.40152272675796392235796567725, −5.26770803610804114861303773339, −3.76060769659553506276694871497, −2.12269688406492296074462119899, 0,
2.12269688406492296074462119899, 3.76060769659553506276694871497, 5.26770803610804114861303773339, 6.40152272675796392235796567725, 7.02864082573576697328686349834, 8.489865387400252022087843283105, 9.056880838216350511563756479245, 10.18616620126561124117671237886, 10.97618644248142866152725079598
