| L(s) = 1 | + 2.49·2-s − 0.710·3-s + 4.24·4-s + 2.06·5-s − 1.77·6-s + 2.11·7-s + 5.60·8-s − 2.49·9-s + 5.15·10-s − 4.92·11-s − 3.01·12-s − 2.71·13-s + 5.29·14-s − 1.46·15-s + 5.52·16-s − 2.73·17-s − 6.23·18-s + 4.78·19-s + 8.75·20-s − 1.50·21-s − 12.2·22-s + 4.00·23-s − 3.98·24-s − 0.744·25-s − 6.79·26-s + 3.90·27-s + 8.99·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.410·3-s + 2.12·4-s + 0.922·5-s − 0.725·6-s + 0.800·7-s + 1.98·8-s − 0.831·9-s + 1.63·10-s − 1.48·11-s − 0.871·12-s − 0.754·13-s + 1.41·14-s − 0.378·15-s + 1.38·16-s − 0.662·17-s − 1.46·18-s + 1.09·19-s + 1.95·20-s − 0.328·21-s − 2.62·22-s + 0.834·23-s − 0.813·24-s − 0.148·25-s − 1.33·26-s + 0.751·27-s + 1.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.476433456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.476433456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 409 | \( 1 - T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 + 0.710T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 - 2.11T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 + 2.71T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 - 4.78T + 19T^{2} \) |
| 23 | \( 1 - 4.00T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 + 6.84T + 31T^{2} \) |
| 37 | \( 1 - 4.49T + 37T^{2} \) |
| 41 | \( 1 - 0.680T + 41T^{2} \) |
| 43 | \( 1 + 8.59T + 43T^{2} \) |
| 47 | \( 1 + 2.83T + 47T^{2} \) |
| 53 | \( 1 + 1.94T + 53T^{2} \) |
| 59 | \( 1 - 5.99T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 + 0.526T + 73T^{2} \) |
| 79 | \( 1 - 3.94T + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 + 2.59T + 89T^{2} \) |
| 97 | \( 1 + 5.67T + 97T^{2} \) |
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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
−11.35487842020961701715859348024, −10.85471252278600700590406998658, −9.720622389741546809432848611747, −8.216296035665493255879957761980, −7.10703296482604848383909433428, −6.02987388187290186957485423300, −5.13582884244668015378821420144, −4.92884972784152255387519235130, −3.06959594826474349291854239106, −2.18269197262669707663212277654,
2.18269197262669707663212277654, 3.06959594826474349291854239106, 4.92884972784152255387519235130, 5.13582884244668015378821420144, 6.02987388187290186957485423300, 7.10703296482604848383909433428, 8.216296035665493255879957761980, 9.720622389741546809432848611747, 10.85471252278600700590406998658, 11.35487842020961701715859348024
