| L(s) = 1 | + 9.64·2-s − 34.8·4-s + 306.·5-s + 733.·7-s − 1.57e3·8-s + 2.96e3·10-s − 3.88e3·11-s − 5.97e3·13-s + 7.07e3·14-s − 1.06e4·16-s − 1.40e4·17-s + 1.10e4·19-s − 1.07e4·20-s − 3.74e4·22-s − 1.21e4·23-s + 1.60e4·25-s − 5.76e4·26-s − 2.55e4·28-s − 116.·29-s − 6.55e4·31-s + 9.79e4·32-s − 1.35e5·34-s + 2.25e5·35-s − 3.82e5·37-s + 1.06e5·38-s − 4.82e5·40-s + 3.70e5·41-s + ⋯ |
| L(s) = 1 | + 0.852·2-s − 0.272·4-s + 1.09·5-s + 0.807·7-s − 1.08·8-s + 0.936·10-s − 0.879·11-s − 0.754·13-s + 0.688·14-s − 0.653·16-s − 0.692·17-s + 0.368·19-s − 0.299·20-s − 0.750·22-s − 0.208·23-s + 0.205·25-s − 0.643·26-s − 0.220·28-s − 0.000889·29-s − 0.395·31-s + 0.528·32-s − 0.590·34-s + 0.887·35-s − 1.24·37-s + 0.314·38-s − 1.19·40-s + 0.839·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 9.64T + 128T^{2} \) |
| 5 | \( 1 - 306.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 733.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.88e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.97e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.40e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.10e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 116.T + 1.72e10T^{2} \) |
| 31 | \( 1 + 6.55e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.70e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.68e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.30e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.45e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.87e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.20e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.04e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.77e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.58e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.20e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.72e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.55e6T + 8.07e13T^{2} \) |
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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
−10.64436143425049006824410298777, −9.661193803035458932408181881850, −8.752895282206212615207565942039, −7.51175735212935787649801596680, −6.07537980315793566553564130816, −5.24711781284460884863322312322, −4.49415137150874207805014125071, −2.90795986183609759987089929033, −1.81138324880307653563914988975, 0,
1.81138324880307653563914988975, 2.90795986183609759987089929033, 4.49415137150874207805014125071, 5.24711781284460884863322312322, 6.07537980315793566553564130816, 7.51175735212935787649801596680, 8.752895282206212615207565942039, 9.661193803035458932408181881850, 10.64436143425049006824410298777
