| L(s) = 1 | + 15.3·2-s + 73.3·3-s + 108.·4-s + 1.12e3·6-s − 700.·7-s − 301.·8-s + 3.18e3·9-s − 2.68e3·11-s + 7.94e3·12-s + 3.01e3·13-s − 1.07e4·14-s − 1.85e4·16-s + 2.92e4·17-s + 4.90e4·18-s − 2.86e4·19-s − 5.13e4·21-s − 4.13e4·22-s − 6.82e4·23-s − 2.21e4·24-s + 4.63e4·26-s + 7.33e4·27-s − 7.58e4·28-s − 1.89e5·29-s − 2.27e5·31-s − 2.45e5·32-s − 1.96e5·33-s + 4.50e5·34-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 1.56·3-s + 0.846·4-s + 2.13·6-s − 0.771·7-s − 0.208·8-s + 1.45·9-s − 0.608·11-s + 1.32·12-s + 0.380·13-s − 1.04·14-s − 1.12·16-s + 1.44·17-s + 1.98·18-s − 0.959·19-s − 1.20·21-s − 0.827·22-s − 1.17·23-s − 0.326·24-s + 0.517·26-s + 0.717·27-s − 0.653·28-s − 1.44·29-s − 1.36·31-s − 1.32·32-s − 0.954·33-s + 1.96·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 15.3T + 128T^{2} \) |
| 3 | \( 1 - 73.3T + 2.18e3T^{2} \) |
| 7 | \( 1 + 700.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.68e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.01e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.92e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.86e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.82e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.89e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.27e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.54e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.40e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.83e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.00e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.32e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.37e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 6.36e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.52e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.03e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.76e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.32e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.19e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.73e5T + 8.07e13T^{2} \) |
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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
−9.075835711586107017875545241615, −8.102248169770386136267708262269, −7.32006264505967437070354204076, −6.14879261898838133210190819312, −5.35665643079692578083310920431, −3.94572440137224883853025381857, −3.60017194916969515652724312594, −2.71541566616299873773324590280, −1.86009665292831009027568730739, 0,
1.86009665292831009027568730739, 2.71541566616299873773324590280, 3.60017194916969515652724312594, 3.94572440137224883853025381857, 5.35665643079692578083310920431, 6.14879261898838133210190819312, 7.32006264505967437070354204076, 8.102248169770386136267708262269, 9.075835711586107017875545241615
