| L(s) = 1 | − 5.46·2-s + 0.820·3-s + 21.8·4-s − 4.48·6-s + 22.0·7-s − 75.4·8-s − 26.3·9-s + 8.31·11-s + 17.9·12-s + 24.7·13-s − 120.·14-s + 237.·16-s − 17·17-s + 143.·18-s + 79.8·19-s + 18.0·21-s − 45.4·22-s + 157.·23-s − 61.9·24-s − 134.·26-s − 43.7·27-s + 480.·28-s − 226.·29-s + 30.3·31-s − 692.·32-s + 6.82·33-s + 92.8·34-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.157·3-s + 2.72·4-s − 0.304·6-s + 1.18·7-s − 3.33·8-s − 0.975·9-s + 0.227·11-s + 0.430·12-s + 0.527·13-s − 2.29·14-s + 3.70·16-s − 0.242·17-s + 1.88·18-s + 0.964·19-s + 0.187·21-s − 0.440·22-s + 1.42·23-s − 0.526·24-s − 1.01·26-s − 0.311·27-s + 3.24·28-s − 1.45·29-s + 0.175·31-s − 3.82·32-s + 0.0360·33-s + 0.468·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9204523566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9204523566\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 5.46T + 8T^{2} \) |
| 3 | \( 1 - 0.820T + 27T^{2} \) |
| 7 | \( 1 - 22.0T + 343T^{2} \) |
| 11 | \( 1 - 8.31T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 79.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 30.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 232.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 384.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 596.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 402.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 299.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 527.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 462.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 335.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 171.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 414.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.31e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 844.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−10.82492970779218712184456695781, −9.550361869387254498043971791713, −8.945876848240728508908817841149, −8.144291497776284171366744664645, −7.53811855125051496624402918834, −6.41681063147727958021617889235, −5.28541925258136655289989303734, −3.21044519005423195608538155287, −1.95916825747549714099418083337, −0.835058222112317193451502669235,
0.835058222112317193451502669235, 1.95916825747549714099418083337, 3.21044519005423195608538155287, 5.28541925258136655289989303734, 6.41681063147727958021617889235, 7.53811855125051496624402918834, 8.144291497776284171366744664645, 8.945876848240728508908817841149, 9.550361869387254498043971791713, 10.82492970779218712184456695781
