| L(s) = 1 | + 4.73·3-s − 18.0·5-s + 0.213·7-s − 4.58·9-s − 3.39·11-s − 90.7·13-s − 85.5·15-s − 2.59·17-s + 19·19-s + 1.01·21-s + 26.6·23-s + 201.·25-s − 149.·27-s + 60.1·29-s − 176.·31-s − 16.0·33-s − 3.85·35-s − 154.·37-s − 429.·39-s + 434.·41-s − 365.·43-s + 82.8·45-s + 204.·47-s − 342.·49-s − 12.2·51-s − 135.·53-s + 61.3·55-s + ⋯ |
| L(s) = 1 | + 0.911·3-s − 1.61·5-s + 0.0115·7-s − 0.169·9-s − 0.0930·11-s − 1.93·13-s − 1.47·15-s − 0.0370·17-s + 0.229·19-s + 0.0104·21-s + 0.241·23-s + 1.61·25-s − 1.06·27-s + 0.384·29-s − 1.02·31-s − 0.0847·33-s − 0.0186·35-s − 0.684·37-s − 1.76·39-s + 1.65·41-s − 1.29·43-s + 0.274·45-s + 0.633·47-s − 0.999·49-s − 0.0337·51-s − 0.351·53-s + 0.150·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 4.73T + 27T^{2} \) |
| 5 | \( 1 + 18.0T + 125T^{2} \) |
| 7 | \( 1 - 0.213T + 343T^{2} \) |
| 11 | \( 1 + 3.39T + 1.33e3T^{2} \) |
| 13 | \( 1 + 90.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.59T + 4.91e3T^{2} \) |
| 23 | \( 1 - 26.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 60.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 135.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 759.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 284.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 590.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 972.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 368.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 204.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 782.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 213.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.02598662687045501501526509585, −11.20933989573261919616816403860, −9.818124951287173822617524905056, −8.712379338474662743522963162737, −7.79599978064815841842736831794, −7.13871577742217804496965110189, −5.05181218860378821487185358045, −3.78771939816173543288843192489, −2.62957810903665442177864465320, 0,
2.62957810903665442177864465320, 3.78771939816173543288843192489, 5.05181218860378821487185358045, 7.13871577742217804496965110189, 7.79599978064815841842736831794, 8.712379338474662743522963162737, 9.818124951287173822617524905056, 11.20933989573261919616816403860, 12.02598662687045501501526509585
