| L(s) = 1 | − 4i·2-s + 9i·3-s − 16·4-s + 36·6-s − 47i·7-s + 64i·8-s − 81·9-s + 222·11-s − 144i·12-s + 101i·13-s − 188·14-s + 256·16-s − 162i·17-s + 324i·18-s − 1.68e3·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.362i·7-s + 0.353i·8-s − 0.333·9-s + 0.553·11-s − 0.288i·12-s + 0.165i·13-s − 0.256·14-s + 0.250·16-s − 0.135i·17-s + 0.235i·18-s − 1.07·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 3 | \( 1 - 9iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 47iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 222T + 1.61e5T^{2} \) |
| 13 | \( 1 - 101iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 162iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.68e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 306iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 7.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.64e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.43e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.09e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.79e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 107iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.47e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.08e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 3.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 823iT - 8.58e9T^{2} \) |
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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
−11.30828547456348051723441514202, −10.65691070981280731556371556599, −9.534182132820765555922583451457, −8.775312055017578442320075923668, −7.33665356036869546603351625517, −5.81270779329370545836516975207, −4.43266760741805201932565400363, −3.48564509307124190856747116958, −1.81955799539616738286809810787, 0,
1.84427172740867241762299622034, 3.70942261995431498736699402934, 5.28436171032639350801293678299, 6.34653990704058382176832654020, 7.30775720150262145999945714229, 8.443776184505810865264754489659, 9.272472037583474456717320333148, 10.67024623534196603513239917434, 11.87947075405782901851008815028
