| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 4·7-s + 8-s + 9-s + 10-s − 12-s − 6·13-s + 4·14-s − 15-s + 16-s + 2·17-s + 18-s + 19-s + 20-s − 4·21-s + 8·23-s − 24-s + 25-s − 6·26-s − 27-s + 4·28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 1.66·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.872·21-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.755·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.26606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26606\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−11.00622336986043507039085651351, −10.07997888437990250281069279781, −9.075848997427652752699530458232, −7.70755210068182572865041945527, −7.19070890117421355121217701033, −5.87466392302761204349542038383, −5.02351044483092250684422865721, −4.55866188918253382737360928969, −2.80522904846532735262522909925, −1.50384411563093130227063831189,
1.50384411563093130227063831189, 2.80522904846532735262522909925, 4.55866188918253382737360928969, 5.02351044483092250684422865721, 5.87466392302761204349542038383, 7.19070890117421355121217701033, 7.70755210068182572865041945527, 9.075848997427652752699530458232, 10.07997888437990250281069279781, 11.00622336986043507039085651351
