| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 3·11-s − 12-s − 2·13-s − 14-s + 16-s + 18-s + 4·19-s + 21-s − 3·22-s − 24-s − 2·26-s − 27-s − 28-s − 3·29-s + 5·31-s + 32-s + 3·33-s + 36-s + 2·37-s + 4·38-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 + 1/3·9-s − 0.904·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.218·21-s − 0.639·22-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s − 0.557·29-s + 0.898·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s + 0.328·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.19061804330521, −13.62130215440575, −13.25683168488837, −12.76656364253829, −12.30966267610149, −11.85242232436148, −11.31612459734899, −10.93296395788937, −10.21584858992792, −9.914891739442405, −9.425934803192366, −8.636627777190951, −7.941642862036241, −7.551084518115220, −6.966718595071042, −6.497849453031464, −5.798961017586862, −5.444843626848659, −4.889639920290506, −4.393248399036849, −3.682371881140086, −3.000998752408790, −2.553458127733656, −1.742928218116620, −0.8752195456053117, 0,
0.8752195456053117, 1.742928218116620, 2.553458127733656, 3.000998752408790, 3.682371881140086, 4.393248399036849, 4.889639920290506, 5.444843626848659, 5.798961017586862, 6.497849453031464, 6.966718595071042, 7.551084518115220, 7.941642862036241, 8.636627777190951, 9.425934803192366, 9.914891739442405, 10.21584858992792, 10.93296395788937, 11.31612459734899, 11.85242232436148, 12.30966267610149, 12.76656364253829, 13.25683168488837, 13.62130215440575, 14.19061804330521
