| L(s) = 1 | + 3-s − 2·5-s − 2·7-s + 9-s + 11-s − 2·13-s − 2·15-s + 6·19-s − 2·21-s − 25-s + 27-s + 8·29-s − 8·31-s + 33-s + 4·35-s − 10·37-s − 2·39-s − 8·41-s + 2·43-s − 2·45-s + 8·47-s − 3·49-s − 2·53-s − 2·55-s + 6·57-s − 12·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.516·15-s + 1.37·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s + 1.48·29-s − 1.43·31-s + 0.174·33-s + 0.676·35-s − 1.64·37-s − 0.320·39-s − 1.24·41-s + 0.304·43-s − 0.298·45-s + 1.16·47-s − 3/7·49-s − 0.274·53-s − 0.269·55-s + 0.794·57-s − 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7373664176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7373664176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.46600058610562, −12.70272657782919, −12.30882307062038, −12.04750441242809, −11.56570670819715, −10.89520546724838, −10.40279252975233, −9.937708182859233, −9.366500838730921, −9.093676494750909, −8.445213707668454, −8.001503194949057, −7.420475669545389, −7.056669601314498, −6.709126426914493, −5.820960484928214, −5.452515525395722, −4.611767151662831, −4.310428340939662, −3.486301684137878, −3.221859231741962, −2.792127670988567, −1.814326907837550, −1.286094546776305, −0.2446007710895278,
0.2446007710895278, 1.286094546776305, 1.814326907837550, 2.792127670988567, 3.221859231741962, 3.486301684137878, 4.310428340939662, 4.611767151662831, 5.452515525395722, 5.820960484928214, 6.709126426914493, 7.056669601314498, 7.420475669545389, 8.001503194949057, 8.445213707668454, 9.093676494750909, 9.366500838730921, 9.937708182859233, 10.40279252975233, 10.89520546724838, 11.56570670819715, 12.04750441242809, 12.30882307062038, 12.70272657782919, 13.46600058610562
