| L(s) = 1 | − 0.836·3-s + 287.·5-s + 124.·7-s − 2.18e3·9-s − 5.42e3·11-s + 3.56e3·13-s − 240.·15-s − 1.04e3·17-s + 5.06e4·19-s − 104.·21-s + 1.48e4·23-s + 4.24e3·25-s + 3.66e3·27-s − 2.43e4·29-s + 1.16e5·31-s + 4.54e3·33-s + 3.56e4·35-s − 6.27e4·37-s − 2.98e3·39-s − 5.67e4·41-s − 6.05e5·43-s − 6.27e5·45-s − 9.87e5·47-s − 8.08e5·49-s + 870.·51-s + 1.67e6·53-s − 1.55e6·55-s + ⋯ |
| L(s) = 1 | − 0.0178·3-s + 1.02·5-s + 0.137·7-s − 0.999·9-s − 1.22·11-s + 0.450·13-s − 0.0183·15-s − 0.0513·17-s + 1.69·19-s − 0.00245·21-s + 0.254·23-s + 0.0543·25-s + 0.0357·27-s − 0.185·29-s + 0.703·31-s + 0.0220·33-s + 0.140·35-s − 0.203·37-s − 0.00806·39-s − 0.128·41-s − 1.16·43-s − 1.02·45-s − 1.38·47-s − 0.981·49-s + 0.000919·51-s + 1.54·53-s − 1.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + 2.43e4T \) |
| good | 3 | \( 1 + 0.836T + 2.18e3T^{2} \) |
| 5 | \( 1 - 287.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 124.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.42e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.56e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.04e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.06e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.48e4T + 3.40e9T^{2} \) |
| 31 | \( 1 - 1.16e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 6.27e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.67e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.05e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.87e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.67e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.60e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.05e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.78e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.67e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 7.19e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.69e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.75e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.13e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.45e7T + 8.07e13T^{2} \) |
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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
−9.555457827442733553231670469053, −8.526302614486343316451069599939, −7.74006694797305546977629991189, −6.50640832477457116525082337826, −5.53927355253043421831432607782, −5.06230927646139818444123331502, −3.32208819220010502424240047449, −2.50343767005735827805204084177, −1.33309671148704508182347767478, 0,
1.33309671148704508182347767478, 2.50343767005735827805204084177, 3.32208819220010502424240047449, 5.06230927646139818444123331502, 5.53927355253043421831432607782, 6.50640832477457116525082337826, 7.74006694797305546977629991189, 8.526302614486343316451069599939, 9.555457827442733553231670469053
