| L(s) = 1 | + 2-s + 3·3-s + 4-s − 5-s + 3·6-s + 8-s + 6·9-s − 10-s − 2·11-s + 3·12-s + 7·13-s − 3·15-s + 16-s − 4·17-s + 6·18-s + 19-s − 20-s − 2·22-s + 3·23-s + 3·24-s + 25-s + 7·26-s + 9·27-s + 9·29-s − 3·30-s − 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s + 0.353·8-s + 2·9-s − 0.316·10-s − 0.603·11-s + 0.866·12-s + 1.94·13-s − 0.774·15-s + 1/4·16-s − 0.970·17-s + 1.41·18-s + 0.229·19-s − 0.223·20-s − 0.426·22-s + 0.625·23-s + 0.612·24-s + 1/5·25-s + 1.37·26-s + 1.73·27-s + 1.67·29-s − 0.547·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.78776788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.78776788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 617 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−12.98703490460288, −12.43691189214771, −11.76263097778840, −11.37039921116267, −10.87090496770150, −10.27558507702841, −10.14650701985108, −9.180071717013379, −8.822450548914188, −8.573095623862880, −8.164297507766800, −7.571601705994856, −7.147778064000476, −6.710877801473166, −6.119375358563173, −5.582671057386894, −4.854666597598288, −4.311664501131702, −4.013703657330667, −3.322202189446680, −3.175354774824498, −2.453861385327850, −2.068370395852065, −1.276038098681667, −0.7392818999794672,
0.7392818999794672, 1.276038098681667, 2.068370395852065, 2.453861385327850, 3.175354774824498, 3.322202189446680, 4.013703657330667, 4.311664501131702, 4.854666597598288, 5.582671057386894, 6.119375358563173, 6.710877801473166, 7.147778064000476, 7.571601705994856, 8.164297507766800, 8.573095623862880, 8.822450548914188, 9.180071717013379, 10.14650701985108, 10.27558507702841, 10.87090496770150, 11.37039921116267, 11.76263097778840, 12.43691189214771, 12.98703490460288
