| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 6·11-s − 4·13-s + 15-s + 2·17-s − 4·19-s − 21-s − 23-s + 25-s − 27-s + 2·29-s + 2·31-s − 6·33-s − 35-s + 12·37-s + 4·39-s + 6·41-s + 4·43-s − 45-s + 4·47-s + 49-s − 2·51-s + 12·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s − 1.04·33-s − 0.169·35-s + 1.97·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 1.64·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.043958314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.043958314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.78952429397725, −14.34540484838277, −14.02446144401406, −13.14361104953473, −12.43716817047628, −12.29275734503877, −11.61818883631325, −11.36058519993039, −10.67973355304904, −10.11576881672027, −9.475099537288646, −9.086902023823640, −8.403825691286544, −7.679612693870683, −7.344881729177260, −6.564262276847663, −6.196726409116552, −5.552085340866546, −4.700679106351250, −4.263865330337454, −3.892950145930599, −2.855916218865118, −2.157269975778352, −1.228265687610311, −0.6139617535033050,
0.6139617535033050, 1.228265687610311, 2.157269975778352, 2.855916218865118, 3.892950145930599, 4.263865330337454, 4.700679106351250, 5.552085340866546, 6.196726409116552, 6.564262276847663, 7.344881729177260, 7.679612693870683, 8.403825691286544, 9.086902023823640, 9.475099537288646, 10.11576881672027, 10.67973355304904, 11.36058519993039, 11.61818883631325, 12.29275734503877, 12.43716817047628, 13.14361104953473, 14.02446144401406, 14.34540484838277, 14.78952429397725
