| L(s) = 1 | − 0.402·2-s + 50.9·3-s − 127.·4-s − 384.·5-s − 20.4·6-s − 88.4·7-s + 102.·8-s + 404.·9-s + 154.·10-s − 1.54e3·11-s − 6.50e3·12-s − 1.07e4·13-s + 35.5·14-s − 1.95e4·15-s + 1.63e4·16-s + 1.33e3·17-s − 162.·18-s + 2.04e3·19-s + 4.91e4·20-s − 4.50e3·21-s + 620.·22-s + 1.21e4·23-s + 5.23e3·24-s + 6.98e4·25-s + 4.30e3·26-s − 9.07e4·27-s + 1.13e4·28-s + ⋯ |
| L(s) = 1 | − 0.0355·2-s + 1.08·3-s − 0.998·4-s − 1.37·5-s − 0.0387·6-s − 0.0974·7-s + 0.0710·8-s + 0.185·9-s + 0.0489·10-s − 0.349·11-s − 1.08·12-s − 1.35·13-s + 0.00346·14-s − 1.49·15-s + 0.996·16-s + 0.0658·17-s − 0.00658·18-s + 0.0684·19-s + 1.37·20-s − 0.106·21-s + 0.0124·22-s + 0.208·23-s + 0.0773·24-s + 0.894·25-s + 0.0480·26-s − 0.887·27-s + 0.0973·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 + 0.402T + 128T^{2} \) |
| 3 | \( 1 - 50.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 384.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 88.4T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.07e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.33e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.04e3T + 8.93e8T^{2} \) |
| 29 | \( 1 - 8.36e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.48e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.69e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.43e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.85e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.97e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.35e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.47e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.41e7T + 8.07e13T^{2} \) |
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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
−15.26465499638114515117145314045, −14.49521149274910143064516688385, −13.19642673757745437351184968179, −11.86836316409778932434866738110, −9.855154061845855426011124582736, −8.501134136330902748433768056869, −7.63694394624244850082204076734, −4.64307867786491467065333592975, −3.16804945304163144511176811500, 0,
3.16804945304163144511176811500, 4.64307867786491467065333592975, 7.63694394624244850082204076734, 8.501134136330902748433768056869, 9.855154061845855426011124582736, 11.86836316409778932434866738110, 13.19642673757745437351184968179, 14.49521149274910143064516688385, 15.26465499638114515117145314045
