| L(s) = 1 | + (−4.66 + 2.69i)2-s + (15.4 + 8.89i)3-s + (−1.49 + 2.58i)4-s + (55.0 − 9.76i)5-s − 95.8·6-s + (79.8 + 102. i)7-s − 188. i·8-s + (36.7 + 63.7i)9-s + (−230. + 193. i)10-s + (−268. + 465. i)11-s + (−45.9 + 26.5i)12-s + 751. i·13-s + (−647. − 261. i)14-s + (935. + 339. i)15-s + (459. + 796. i)16-s + (−2.72 − 1.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.824 + 0.476i)2-s + (0.988 + 0.570i)3-s + (−0.0466 + 0.0807i)4-s + (0.984 − 0.174i)5-s − 1.08·6-s + (0.615 + 0.787i)7-s − 1.04i·8-s + (0.151 + 0.262i)9-s + (−0.728 + 0.612i)10-s + (−0.669 + 1.16i)11-s + (−0.0921 + 0.0532i)12-s + 1.23i·13-s + (−0.882 − 0.356i)14-s + (1.07 + 0.389i)15-s + (0.449 + 0.777i)16-s + (−0.00228 − 0.00132i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.917319 + 1.18358i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.917319 + 1.18358i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-55.0 + 9.76i)T \) |
| 7 | \( 1 + (-79.8 - 102. i)T \) |
| good | 2 | \( 1 + (4.66 - 2.69i)T + (16 - 27.7i)T^{2} \) |
| 3 | \( 1 + (-15.4 - 8.89i)T + (121.5 + 210. i)T^{2} \) |
| 11 | \( 1 + (268. - 465. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 751. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (2.72 + 1.57i)T + (7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.15e3 + 1.99e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.27e3 + 735. i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 6.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (2.37e3 - 4.11e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.25e3 - 724. i)T + (3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 2.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.67e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.97e4 + 1.14e4i)T + (1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.25e4 - 7.24e3i)T + (2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.71e4 + 2.96e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.41e3 + 4.18e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (6.66e3 + 3.84e3i)T + (6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-5.23e3 - 3.02e3i)T + (1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (4.57e4 + 7.92e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 - 2.99e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + (-1.92e3 - 3.34e3i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 3.60e4iT - 8.58e9T^{2} \) |
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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
−15.79199538528788579524827170898, −14.91602254908537459337597026570, −13.72533947153533717107360557893, −12.37304770650958406954536892802, −10.23915837540896646433921427316, −9.082394678405393792070168401492, −8.660039509826856068022750510987, −6.89469042224485374855183019778, −4.67385470568943624122584583322, −2.32495401564742046288054715079,
1.16955240182241670741393533770, 2.69539500487306891486350813674, 5.61452285464696424662095003256, 7.84439521051708446146027596306, 8.621305895310088445778469438484, 10.18014255134551715820254043861, 10.89469743219972058050911449606, 13.10785084482888191145563299719, 13.93504568657211576898868173398, 14.71788563450302979203263049445
