| L(s) = 1 | − 6.16·5-s − 343·7-s + 2.33e3·11-s + 1.44e3·13-s − 3.20e4·17-s + 2.95e4·19-s + 7.95e4·23-s − 7.80e4·25-s + 3.19e4·29-s + 6.28e4·31-s + 2.11e3·35-s + 1.14e5·37-s − 8.42e5·41-s − 8.01e5·43-s + 7.04e5·47-s + 1.17e5·49-s + 1.83e6·53-s − 1.43e4·55-s − 2.29e6·59-s + 2.90e5·61-s − 8.90e3·65-s − 6.00e5·67-s − 3.81e5·71-s − 3.50e6·73-s − 8.01e5·77-s − 5.03e6·79-s + 5.75e6·83-s + ⋯ |
| L(s) = 1 | − 0.0220·5-s − 0.377·7-s + 0.529·11-s + 0.182·13-s − 1.58·17-s + 0.988·19-s + 1.36·23-s − 0.999·25-s + 0.243·29-s + 0.378·31-s + 0.00833·35-s + 0.371·37-s − 1.90·41-s − 1.53·43-s + 0.990·47-s + 0.142·49-s + 1.69·53-s − 0.0116·55-s − 1.45·59-s + 0.163·61-s − 0.00402·65-s − 0.243·67-s − 0.126·71-s − 1.05·73-s − 0.200·77-s − 1.14·79-s + 1.10·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 343T \) |
| good | 5 | \( 1 + 6.16T + 7.81e4T^{2} \) |
| 11 | \( 1 - 2.33e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.95e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.95e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.19e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.28e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.14e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.42e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.01e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.04e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.83e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.29e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.90e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.00e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.81e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.50e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.03e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.75e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.09e7T + 8.07e13T^{2} \) |
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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
−10.29450630012410914798887427177, −9.296684193186107683877607273151, −8.525707031837782686499651974025, −7.18871905068514839364563740478, −6.42302130216767427645239305831, −5.16340727838930836059552615389, −3.98932921317327353526885517996, −2.80851202304802577507559877586, −1.39743072023695782493108300958, 0,
1.39743072023695782493108300958, 2.80851202304802577507559877586, 3.98932921317327353526885517996, 5.16340727838930836059552615389, 6.42302130216767427645239305831, 7.18871905068514839364563740478, 8.525707031837782686499651974025, 9.296684193186107683877607273151, 10.29450630012410914798887427177
