| L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.128 − 0.281i)3-s + (−0.959 + 0.281i)4-s + (−1.69 + 1.95i)5-s + (−0.296 − 0.0871i)6-s + (−0.841 + 0.540i)7-s + (0.415 + 0.909i)8-s + (1.90 + 2.19i)9-s + (2.17 + 1.39i)10-s + (0.223 − 1.55i)11-s + (−0.0440 + 0.306i)12-s + (5.87 + 3.77i)13-s + (0.654 + 0.755i)14-s + (0.332 + 0.727i)15-s + (0.841 − 0.540i)16-s + (3.24 + 0.953i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.0742 − 0.162i)3-s + (−0.479 + 0.140i)4-s + (−0.756 + 0.872i)5-s + (−0.121 − 0.0355i)6-s + (−0.317 + 0.204i)7-s + (0.146 + 0.321i)8-s + (0.633 + 0.731i)9-s + (0.687 + 0.441i)10-s + (0.0673 − 0.468i)11-s + (−0.0127 + 0.0884i)12-s + (1.62 + 1.04i)13-s + (0.175 + 0.201i)14-s + (0.0857 + 0.187i)15-s + (0.210 − 0.135i)16-s + (0.787 + 0.231i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03464 + 0.217269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03464 + 0.217269i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (3.55 - 3.22i)T \) |
| good | 3 | \( 1 + (-0.128 + 0.281i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (1.69 - 1.95i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.223 + 1.55i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-5.87 - 3.77i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.24 - 0.953i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (2.78 - 0.818i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (4.89 + 1.43i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.08 - 4.56i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (1.70 + 1.96i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.81 + 3.24i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.433 + 0.948i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 4.35T + 47T^{2} \) |
| 53 | \( 1 + (-3.30 + 2.12i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-5.95 - 3.82i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.26 - 7.15i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (1.24 + 8.67i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.40 + 9.75i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-4.37 + 1.28i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (0.936 + 0.601i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (5.51 + 6.36i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (5.26 - 11.5i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-7.73 + 8.92i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.47959601206557650153536088019, −10.90961614956879220638495153518, −10.08312511795763236852282177965, −8.868400785417086116136646204735, −7.986924420267179779648374217798, −6.97338808520697003650017481111, −5.83245612145959815550730587010, −4.09845852692675787873409445705, −3.38688714897721302240782421594, −1.76857923910478979730266992533,
0.868499222069433642100972381409, 3.62904239701803446654474066103, 4.38310542151164936687500037301, 5.72402154033278344222555715100, 6.73169733066998596518232807080, 7.922789119573308295899457005159, 8.524438051047400484073003080253, 9.559374774249686954320147867422, 10.42044518971033009845435958315, 11.67837252201584852909210929164
