| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s − 11-s + 12-s + 4·13-s + 15-s + 16-s − 2·17-s − 18-s − 19-s + 20-s + 22-s + 2·23-s − 24-s − 4·25-s − 4·26-s + 27-s + 5·29-s − 30-s − 5·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-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 + 0.213·22-s + 0.417·23-s − 0.204·24-s − 4/5·25-s − 0.784·26-s + 0.192·27-s + 0.928·29-s − 0.182·30-s − 0.898·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.990134787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990134787\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.376185632327656980272796793199, −7.46494418982543420214511982406, −6.94070417037265053821475603891, −6.04650157533589038058500402145, −5.49277700071690596840528303475, −4.31075633122278854614460570418, −3.56900159281645566028443639506, −2.57948140705267200611610013457, −1.89383435739567796195650240633, −0.826179676376123662430802353520,
0.826179676376123662430802353520, 1.89383435739567796195650240633, 2.57948140705267200611610013457, 3.56900159281645566028443639506, 4.31075633122278854614460570418, 5.49277700071690596840528303475, 6.04650157533589038058500402145, 6.94070417037265053821475603891, 7.46494418982543420214511982406, 8.376185632327656980272796793199
