| L(s) = 1 | + (−0.819 − 0.573i)2-s + (−1.56 + 0.749i)3-s + (0.342 + 0.939i)4-s + (2.02 − 0.941i)5-s + (1.70 + 0.281i)6-s + (5.01 − 1.34i)7-s + (0.258 − 0.965i)8-s + (1.87 − 2.34i)9-s + (−2.20 − 0.392i)10-s + (−1.73 + 1.00i)11-s + (−1.23 − 1.21i)12-s + (−1.02 + 0.0896i)13-s + (−4.88 − 1.77i)14-s + (−2.46 + 2.99i)15-s + (−0.766 + 0.642i)16-s + (0.388 + 0.272i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 − 0.405i)2-s + (−0.901 + 0.432i)3-s + (0.171 + 0.469i)4-s + (0.907 − 0.421i)5-s + (0.697 + 0.115i)6-s + (1.89 − 0.508i)7-s + (0.0915 − 0.341i)8-s + (0.625 − 0.780i)9-s + (−0.696 − 0.124i)10-s + (−0.524 + 0.302i)11-s + (−0.357 − 0.349i)12-s + (−0.284 + 0.0248i)13-s + (−1.30 − 0.474i)14-s + (−0.635 + 0.772i)15-s + (−0.191 + 0.160i)16-s + (0.0942 + 0.0659i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07719 - 0.386363i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07719 - 0.386363i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.819 + 0.573i)T \) |
| 3 | \( 1 + (1.56 - 0.749i)T \) |
| 5 | \( 1 + (-2.02 + 0.941i)T \) |
| 19 | \( 1 + (4.32 + 0.578i)T \) |
| good | 7 | \( 1 + (-5.01 + 1.34i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1.73 - 1.00i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.02 - 0.0896i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.388 - 0.272i)T + (5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (-5.18 - 2.41i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.24 + 7.06i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.75 + 4.76i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.06 - 2.06i)T + 37iT^{2} \) |
| 41 | \( 1 + (4.35 + 5.19i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (4.07 - 1.90i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-4.58 - 6.55i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-2.69 - 1.25i)T + (34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-1.86 - 10.5i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (7.08 - 2.57i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-11.1 + 7.80i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (3.08 - 8.47i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.271 + 3.10i)T + (-71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (4.64 + 5.53i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.322 - 0.0864i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-5.13 - 4.30i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.88 + 4.12i)T + (-33.1 - 91.1i)T^{2} \) |
| 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
−10.58929132807017818298937794957, −10.03365258194003224211661660695, −9.032129301415798811469780829535, −8.106241663087090308199697177501, −7.17452369018221808119579004021, −5.90762292280125085000904286901, −4.90048830264084186712350590368, −4.32832532154231503630795094518, −2.22318770272253616765072671685, −1.05801127821729469186822477378,
1.38514001989258606692847261427, 2.37695284740573607895057686373, 4.97075371866219133170410440690, 5.23159490340484296520946638965, 6.37370292747867741012301767068, 7.17744302202182493960795957970, 8.190925557222968858228105337755, 8.850850279455631818423415951565, 10.26079259460668928798896510744, 10.76993685954004533257505504828
