| L(s) = 1 | + (−0.984 + 0.173i)2-s + (−1.71 + 0.233i)3-s + (0.939 − 0.342i)4-s + (−0.0848 − 0.0712i)5-s + (1.64 − 0.527i)6-s + (−1.74 − 1.99i)7-s + (−0.866 + 0.5i)8-s + (2.89 − 0.801i)9-s + (0.0959 + 0.0554i)10-s + (3.36 + 4.00i)11-s + (−1.53 + 0.806i)12-s + (−3.05 − 0.538i)13-s + (2.06 + 1.65i)14-s + (0.162 + 0.102i)15-s + (0.766 − 0.642i)16-s + (0.286 − 0.497i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.990 + 0.134i)3-s + (0.469 − 0.171i)4-s + (−0.0379 − 0.0318i)5-s + (0.673 − 0.215i)6-s + (−0.658 − 0.752i)7-s + (−0.306 + 0.176i)8-s + (0.963 − 0.267i)9-s + (0.0303 + 0.0175i)10-s + (1.01 + 1.20i)11-s + (−0.442 + 0.232i)12-s + (−0.847 − 0.149i)13-s + (0.550 + 0.443i)14-s + (0.0419 + 0.0264i)15-s + (0.191 − 0.160i)16-s + (0.0696 − 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.100519 + 0.260234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100519 + 0.260234i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (1.71 - 0.233i)T \) |
| 7 | \( 1 + (1.74 + 1.99i)T \) |
| good | 5 | \( 1 + (0.0848 + 0.0712i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-3.36 - 4.00i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.05 + 0.538i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.286 + 0.497i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (7.21 - 4.16i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.79 - 4.93i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (8.47 - 1.49i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.29 - 6.30i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (0.699 - 1.21i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.875 - 4.96i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (5.33 - 4.47i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (7.79 + 2.83i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 6.95iT - 53T^{2} \) |
| 59 | \( 1 + (-10.2 - 8.58i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-4.38 + 12.0i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.0706 - 0.400i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (6.50 + 3.75i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.2 + 7.08i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.16 + 6.63i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.04 + 5.94i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.86 - 6.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.16 - 6.15i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.66268954599434473549718558497, −10.54267006543113160479660066208, −9.973373577736611977657061233155, −9.300936030997138661110414096759, −7.79685809748888492996745171524, −6.87747034918465057808518279710, −6.33192196979572065040804269633, −4.88746494079764724425625301390, −3.81231594511724885571469093191, −1.64327891429226858285168155641,
0.25930492228593692288998165404, 2.21995931252837023346025383097, 3.87621770754713390923277506956, 5.41367578110174203739026351731, 6.38947575339530969069404683847, 6.98278703972245117944957004221, 8.432913931984127668047221424656, 9.230854092156023795454027945885, 10.09544885656382113516503790993, 11.26099823024237280783856094744
