| L(s) = 1 | − 2.34·2-s + 3·3-s − 2.49·4-s − 15.3·5-s − 7.04·6-s + 10.1·7-s + 24.6·8-s + 9·9-s + 36.1·10-s − 15.0·11-s − 7.47·12-s − 23.7·14-s − 46.1·15-s − 37.8·16-s + 90.8·17-s − 21.1·18-s − 114.·19-s + 38.3·20-s + 30.3·21-s + 35.3·22-s + 75.7·23-s + 73.8·24-s + 112.·25-s + 27·27-s − 25.2·28-s + 214.·29-s + 108.·30-s + ⋯ |
| L(s) = 1 | − 0.829·2-s + 0.577·3-s − 0.311·4-s − 1.37·5-s − 0.479·6-s + 0.547·7-s + 1.08·8-s + 0.333·9-s + 1.14·10-s − 0.412·11-s − 0.179·12-s − 0.453·14-s − 0.795·15-s − 0.591·16-s + 1.29·17-s − 0.276·18-s − 1.38·19-s + 0.428·20-s + 0.315·21-s + 0.342·22-s + 0.686·23-s + 0.628·24-s + 0.897·25-s + 0.192·27-s − 0.170·28-s + 1.37·29-s + 0.659·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.34T + 8T^{2} \) |
| 5 | \( 1 + 15.3T + 125T^{2} \) |
| 7 | \( 1 - 10.1T + 343T^{2} \) |
| 11 | \( 1 + 15.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 90.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 284.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 358.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 296.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 316.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 254.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 935.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 240.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 947.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 430.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 496.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 979.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 553.T + 9.12e5T^{2} \) |
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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
−10.14831092475841799606375156338, −8.810482304865378117154100269786, −8.275293458692667804101950265075, −7.80801832707127741347145780308, −6.82658749436413270091609591412, −4.98158075863797044703669918718, −4.23683336992381572977028875538, −3.08057219316364011108262262170, −1.34468235926514046784362854010, 0,
1.34468235926514046784362854010, 3.08057219316364011108262262170, 4.23683336992381572977028875538, 4.98158075863797044703669918718, 6.82658749436413270091609591412, 7.80801832707127741347145780308, 8.275293458692667804101950265075, 8.810482304865378117154100269786, 10.14831092475841799606375156338
