| L(s) = 1 | + (−0.194 − 1.40i)2-s + (−0.987 − 0.156i)3-s + (−1.92 + 0.545i)4-s + (1.56 + 1.60i)5-s + (−0.0269 + 1.41i)6-s + (1.26 − 1.26i)7-s + (1.13 + 2.58i)8-s + (0.951 + 0.309i)9-s + (1.93 − 2.49i)10-s + (2.45 − 0.797i)11-s + (1.98 − 0.237i)12-s + (0.719 + 1.41i)13-s + (−2.01 − 1.52i)14-s + (−1.29 − 1.82i)15-s + (3.40 − 2.09i)16-s + (−0.968 − 6.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.137 − 0.990i)2-s + (−0.570 − 0.0903i)3-s + (−0.962 + 0.272i)4-s + (0.698 + 0.715i)5-s + (−0.0110 + 0.577i)6-s + (0.478 − 0.478i)7-s + (0.402 + 0.915i)8-s + (0.317 + 0.103i)9-s + (0.613 − 0.789i)10-s + (0.739 − 0.240i)11-s + (0.573 − 0.0685i)12-s + (0.199 + 0.391i)13-s + (−0.539 − 0.407i)14-s + (−0.333 − 0.471i)15-s + (0.851 − 0.524i)16-s + (−0.234 − 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968704 - 0.594000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968704 - 0.594000i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.194 + 1.40i)T \) |
| 3 | \( 1 + (0.987 + 0.156i)T \) |
| 5 | \( 1 + (-1.56 - 1.60i)T \) |
| good | 7 | \( 1 + (-1.26 + 1.26i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.45 + 0.797i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.719 - 1.41i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.968 + 6.11i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-4.18 - 3.04i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.77 + 5.44i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (2.10 + 2.90i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (3.46 - 4.77i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-8.50 + 4.33i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (2.24 - 6.91i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.96 - 1.96i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.00589 + 0.0372i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (1.13 - 7.17i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-2.91 + 8.96i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.911 + 2.80i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (6.72 - 1.06i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-1.78 - 2.45i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.68 - 3.40i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (3.75 - 2.73i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.32 + 8.36i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (2.83 - 0.922i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (12.1 + 1.91i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.27899198286336242526398578817, −10.96358969002772608750737566194, −9.803872232757935167016777240279, −9.196182785117587830906315149833, −7.72759409572262029811562364439, −6.66564000105726373980552577927, −5.40292797718699261607971951425, −4.25153894410684193739227608486, −2.83131416887528572718167531552, −1.28406943902850731168573849526,
1.39720537548848555145048442703, 4.05133718800432703021060123186, 5.26686136997753891469383842389, 5.78931568127255048656889527748, 6.90631893511920745336659484028, 8.110783671192870490230126748665, 9.068767931285522597793398170490, 9.692149781594454677617288627854, 10.88516097612894300199878620708, 12.02077835879567025528682714532
