| L(s) = 1 | + 6.38·2-s − 33.9·3-s − 87.2·4-s − 216.·6-s − 107.·7-s − 1.37e3·8-s − 1.03e3·9-s − 8.50e3·11-s + 2.96e3·12-s + 7.81e3·13-s − 686.·14-s + 2.40e3·16-s − 1.76e4·17-s − 6.60e3·18-s + 3.88e4·19-s + 3.64e3·21-s − 5.43e4·22-s + 9.97e4·23-s + 4.66e4·24-s + 4.98e4·26-s + 1.09e5·27-s + 9.38e3·28-s − 1.94e5·29-s − 4.84e3·31-s + 1.91e5·32-s + 2.88e5·33-s − 1.12e5·34-s + ⋯ |
| L(s) = 1 | + 0.564·2-s − 0.725·3-s − 0.681·4-s − 0.409·6-s − 0.118·7-s − 0.948·8-s − 0.473·9-s − 1.92·11-s + 0.494·12-s + 0.986·13-s − 0.0668·14-s + 0.146·16-s − 0.870·17-s − 0.267·18-s + 1.30·19-s + 0.0859·21-s − 1.08·22-s + 1.70·23-s + 0.688·24-s + 0.556·26-s + 1.06·27-s + 0.0807·28-s − 1.47·29-s − 0.0292·31-s + 1.03·32-s + 1.39·33-s − 0.491·34-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 | \( 1 - 6.38T + 128T^{2} \) |
| 3 | \( 1 + 33.9T + 2.18e3T^{2} \) |
| 7 | \( 1 + 107.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 8.50e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.81e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.76e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.88e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.97e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.94e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.84e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.57e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.04e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.72e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.29e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.32e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.98e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.27e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.41e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.57e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.43e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.68e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.39e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.12e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.30e4T + 8.07e13T^{2} \) |
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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
−9.023247923505416924011178595460, −8.256475930340674930200671084157, −7.18196367795822657991893260327, −5.93403175055327252708484260118, −5.38879758623737371246345719577, −4.74828493855331283309415640451, −3.44732405873749180732630261451, −2.64220148136841349007544424474, −0.855221367728428605240564687293, 0,
0.855221367728428605240564687293, 2.64220148136841349007544424474, 3.44732405873749180732630261451, 4.74828493855331283309415640451, 5.38879758623737371246345719577, 5.93403175055327252708484260118, 7.18196367795822657991893260327, 8.256475930340674930200671084157, 9.023247923505416924011178595460
