L(s) = 1 | + (1.33 − 0.462i)2-s + (1.57 − 1.23i)4-s + (3.39 − 0.598i)5-s + (−1.88 + 2.24i)7-s + (1.52 − 2.38i)8-s + (4.25 − 2.37i)10-s + (−0.453 + 2.57i)11-s + (−5.09 − 1.85i)13-s + (−1.47 + 3.87i)14-s + (0.938 − 3.88i)16-s + (−1.15 + 0.667i)17-s + (−0.0790 − 0.0456i)19-s + (4.59 − 5.13i)20-s + (0.585 + 3.65i)22-s + (0.650 − 0.545i)23-s + ⋯ |
L(s) = 1 | + (0.944 − 0.327i)2-s + (0.785 − 0.618i)4-s + (1.51 − 0.267i)5-s + (−0.711 + 0.848i)7-s + (0.539 − 0.841i)8-s + (1.34 − 0.749i)10-s + (−0.136 + 0.776i)11-s + (−1.41 − 0.514i)13-s + (−0.394 + 1.03i)14-s + (0.234 − 0.972i)16-s + (−0.280 + 0.161i)17-s + (−0.0181 − 0.0104i)19-s + (1.02 − 1.14i)20-s + (0.124 + 0.778i)22-s + (0.135 − 0.113i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.43791 - 0.641225i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.43791 - 0.641225i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.33 + 0.462i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-3.39 + 0.598i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.88 - 2.24i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.453 - 2.57i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (5.09 + 1.85i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.15 - 0.667i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0790 + 0.0456i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.650 + 0.545i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.171 - 0.470i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.28 + 5.10i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.75 - 4.77i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.976 + 2.68i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (5.59 + 0.986i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.06 - 4.24i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 7.37iT - 53T^{2} \) |
| 59 | \( 1 + (0.528 + 2.99i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.85 + 1.55i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.64 - 4.53i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (5.91 + 10.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.83 - 10.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.576 - 1.58i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.02 + 0.738i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-12.5 - 7.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.14 + 6.49i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.87653712120406678509876867943, −10.47620806152002295024748569055, −9.795703266087568813504832669675, −9.189279946122172271617673932866, −7.38725240758187976525718357510, −6.26673050698427244218578335646, −5.55144634590761923113893495333, −4.67032818931246969501070968018, −2.81775385780110127724525830726, −2.02960708675828924660291846429,
2.17917591920559008201984475575, 3.34694798224735136095397689259, 4.80466250777227761525100972469, 5.81078217154853992220622477716, 6.68551944117714986542092683346, 7.40059158079464326518781470184, 8.988428615508013032559752433393, 10.00389895849378419704460344576, 10.70706684825127857785531863549, 11.86171953416049106523600000318