| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 4·11-s + 12-s − 2·13-s + 16-s + 2·17-s − 18-s − 4·19-s + 4·22-s + 8·23-s − 24-s + 2·26-s + 27-s − 2·29-s − 32-s − 4·33-s − 2·34-s + 36-s − 6·37-s + 4·38-s − 2·39-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.371·29-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s − 0.986·37-s + 0.648·38-s − 0.320·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.58468109684223834762507309020, −7.19583350968754147849175706324, −6.33730310419803136630278345980, −5.41872877403970531188971220627, −4.80487139453824320621399861149, −3.77514220204483433617290311660, −2.83112751939976817677191821680, −2.37093527222532181233334963895, −1.25870005842989575382527148936, 0,
1.25870005842989575382527148936, 2.37093527222532181233334963895, 2.83112751939976817677191821680, 3.77514220204483433617290311660, 4.80487139453824320621399861149, 5.41872877403970531188971220627, 6.33730310419803136630278345980, 7.19583350968754147849175706324, 7.58468109684223834762507309020
