| L(s) = 1 | − 3.18·2-s − 3·3-s + 2.13·4-s + 9.54·6-s − 7·7-s + 18.6·8-s + 9·9-s − 68.7·11-s − 6.39·12-s + 56.2·13-s + 22.2·14-s − 76.5·16-s + 37.9·17-s − 28.6·18-s − 26.0·19-s + 21·21-s + 218.·22-s + 25.5·23-s − 56.0·24-s − 178.·26-s − 27·27-s − 14.9·28-s + 148.·29-s + 75.7·31-s + 94.0·32-s + 206.·33-s − 120.·34-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 0.577·3-s + 0.266·4-s + 0.649·6-s − 0.377·7-s + 0.825·8-s + 0.333·9-s − 1.88·11-s − 0.153·12-s + 1.19·13-s + 0.425·14-s − 1.19·16-s + 0.541·17-s − 0.375·18-s − 0.314·19-s + 0.218·21-s + 2.11·22-s + 0.231·23-s − 0.476·24-s − 1.34·26-s − 0.192·27-s − 0.100·28-s + 0.949·29-s + 0.439·31-s + 0.519·32-s + 1.08·33-s − 0.609·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 3.18T + 8T^{2} \) |
| 11 | \( 1 + 68.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 25.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 345.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 287.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 528.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 361.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 705.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 393.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 591.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 668.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 295.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 916.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 736.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 142.T + 9.12e5T^{2} \) |
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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
−10.14966253408606047069814360401, −9.160302123370904939949244886400, −8.182934395269455658532732879819, −7.62051225820360759747850318434, −6.42046059536286156303156026858, −5.42686173592035330697151182468, −4.34177921118226258136222589299, −2.76267096956193143699422934381, −1.14556117375897894686527891753, 0,
1.14556117375897894686527891753, 2.76267096956193143699422934381, 4.34177921118226258136222589299, 5.42686173592035330697151182468, 6.42046059536286156303156026858, 7.62051225820360759747850318434, 8.182934395269455658532732879819, 9.160302123370904939949244886400, 10.14966253408606047069814360401
