| L(s) = 1 | + 7·2-s + 11·3-s + 17·4-s + 77·6-s + 197·7-s − 105·8-s − 122·9-s − 468·11-s + 187·12-s + 921·13-s + 1.37e3·14-s − 1.27e3·16-s + 1.10e3·17-s − 854·18-s + 361·19-s + 2.16e3·21-s − 3.27e3·22-s + 3.64e3·23-s − 1.15e3·24-s + 6.44e3·26-s − 4.01e3·27-s + 3.34e3·28-s + 7.52e3·29-s + 1.42e3·31-s − 5.59e3·32-s − 5.14e3·33-s + 7.74e3·34-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.705·3-s + 0.531·4-s + 0.873·6-s + 1.51·7-s − 0.580·8-s − 0.502·9-s − 1.16·11-s + 0.374·12-s + 1.51·13-s + 1.88·14-s − 1.24·16-s + 0.929·17-s − 0.621·18-s + 0.229·19-s + 1.07·21-s − 1.44·22-s + 1.43·23-s − 0.409·24-s + 1.87·26-s − 1.05·27-s + 0.807·28-s + 1.66·29-s + 0.265·31-s − 0.965·32-s − 0.822·33-s + 1.14·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.074029967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.074029967\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 - 7 T + p^{5} T^{2} \) |
| 3 | \( 1 - 11 T + p^{5} T^{2} \) |
| 7 | \( 1 - 197 T + p^{5} T^{2} \) |
| 11 | \( 1 + 468 T + p^{5} T^{2} \) |
| 13 | \( 1 - 921 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1107 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3641 T + p^{5} T^{2} \) |
| 29 | \( 1 - 7525 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1422 T + p^{5} T^{2} \) |
| 37 | \( 1 - 11282 T + p^{5} T^{2} \) |
| 41 | \( 1 + 678 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5974 T + p^{5} T^{2} \) |
| 47 | \( 1 - 11072 T + p^{5} T^{2} \) |
| 53 | \( 1 - 17461 T + p^{5} T^{2} \) |
| 59 | \( 1 + 46305 T + p^{5} T^{2} \) |
| 61 | \( 1 - 16292 T + p^{5} T^{2} \) |
| 67 | \( 1 + 36373 T + p^{5} T^{2} \) |
| 71 | \( 1 + 82208 T + p^{5} T^{2} \) |
| 73 | \( 1 - 43861 T + p^{5} T^{2} \) |
| 79 | \( 1 + 30130 T + p^{5} T^{2} \) |
| 83 | \( 1 - 91626 T + p^{5} T^{2} \) |
| 89 | \( 1 - 79170 T + p^{5} T^{2} \) |
| 97 | \( 1 + 128718 T + p^{5} T^{2} \) |
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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
−10.51112967558247513373588919102, −8.999277094674896054091501245289, −8.358223250387405484647754944374, −7.63490211236736879458588322466, −6.10484932976164333646195212573, −5.28128394215012048854442146875, −4.53309122392308457187306933915, −3.31946864949369114536833123650, −2.57964612064087253454405669060, −1.08321807040080984653596810933,
1.08321807040080984653596810933, 2.57964612064087253454405669060, 3.31946864949369114536833123650, 4.53309122392308457187306933915, 5.28128394215012048854442146875, 6.10484932976164333646195212573, 7.63490211236736879458588322466, 8.358223250387405484647754944374, 8.999277094674896054091501245289, 10.51112967558247513373588919102
