| L(s) = 1 | + 8·2-s + 70.5·3-s + 64·4-s − 243.·5-s + 564.·6-s + 620.·7-s + 512·8-s + 2.79e3·9-s − 1.95e3·10-s − 5.12e3·11-s + 4.51e3·12-s − 8.49e3·13-s + 4.96e3·14-s − 1.72e4·15-s + 4.09e3·16-s − 3.59e4·17-s + 2.23e4·18-s + 4.69e4·19-s − 1.56e4·20-s + 4.37e4·21-s − 4.10e4·22-s + 5.24e4·23-s + 3.61e4·24-s − 1.86e4·25-s − 6.79e4·26-s + 4.26e4·27-s + 3.97e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.50·3-s + 0.5·4-s − 0.872·5-s + 1.06·6-s + 0.683·7-s + 0.353·8-s + 1.27·9-s − 0.616·10-s − 1.16·11-s + 0.754·12-s − 1.07·13-s + 0.483·14-s − 1.31·15-s + 0.250·16-s − 1.77·17-s + 0.902·18-s + 1.57·19-s − 0.436·20-s + 1.03·21-s − 0.821·22-s + 0.898·23-s + 0.533·24-s − 0.239·25-s − 0.758·26-s + 0.417·27-s + 0.341·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 - 70.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 243.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 620.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.12e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.49e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.69e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.24e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.60e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.17e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.76e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.47e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.35e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.36e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.09e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.00e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.42e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.09e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.03e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.18e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.68e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.07e6T + 8.07e13T^{2} \) |
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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
−9.073984827298302609742066571022, −8.075910888262557975087845625157, −7.68948848154390743593241102706, −6.82053553856825229717268021827, −5.04042756085922325323650543100, −4.56298799677468563445307215745, −3.30516786757962239324987648422, −2.70891435432081257857649709357, −1.72380756834874292039824640027, 0,
1.72380756834874292039824640027, 2.70891435432081257857649709357, 3.30516786757962239324987648422, 4.56298799677468563445307215745, 5.04042756085922325323650543100, 6.82053553856825229717268021827, 7.68948848154390743593241102706, 8.075910888262557975087845625157, 9.073984827298302609742066571022
