| L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (−0.841 − 0.540i)6-s + (−0.113 + 0.249i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (1.96 + 0.576i)11-s + (0.959 + 0.281i)12-s + (−1.58 − 3.46i)13-s + (0.0390 − 0.271i)14-s + (0.654 − 0.755i)15-s + (0.415 − 0.909i)16-s + (5.21 + 3.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.0636 − 0.442i)5-s + (−0.343 − 0.220i)6-s + (−0.0430 + 0.0942i)7-s + (−0.231 + 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.131 + 0.287i)10-s + (0.592 + 0.173i)11-s + (0.276 + 0.0813i)12-s + (−0.438 − 0.960i)13-s + (0.0104 − 0.0725i)14-s + (0.169 − 0.195i)15-s + (0.103 − 0.227i)16-s + (1.26 + 0.813i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26866 + 0.287172i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26866 + 0.287172i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-0.871 - 4.71i)T \) |
| good | 7 | \( 1 + (0.113 - 0.249i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.96 - 0.576i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.58 + 3.46i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.21 - 3.35i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.93 + 1.24i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-6.11 - 3.93i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.32 + 4.98i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.53 + 10.6i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (1.31 + 9.15i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-7.01 - 8.09i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 9.93T + 47T^{2} \) |
| 53 | \( 1 + (3.39 - 7.43i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-4.77 - 10.4i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.404 + 0.466i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-3.00 + 0.883i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-15.6 + 4.60i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (5.48 - 3.52i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.75 + 3.83i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.09 - 7.59i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (10.2 + 11.8i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.334 - 2.32i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.25855858579594393561815755802, −9.605337418541377931600291162943, −8.898318537960520397985446901971, −7.955901321017699722377413669272, −7.39392989316437880124695293487, −6.00802503769640459483036574302, −5.21698620485354144723613066869, −3.93527957779356527611528369441, −2.77252103944853735046007458088, −1.15606098920338638604857964165,
1.10484603662317498188186419898, 2.53421896432472704199997534818, 3.47808726798332512406495961719, 4.86295539112327725430809400743, 6.46074445647410624649642132669, 6.87498217183438281441511842687, 7.953583923207968052175199025031, 8.578896568513225332189212096336, 9.777145780261109065876416845039, 10.01220529749781822426270604269
