| L(s) = 1 | + (1.82 − 0.0367i)2-s + (0.0575 + 0.161i)3-s + (1.33 − 0.0539i)4-s + (0.217 − 2.15i)5-s + (0.111 + 0.292i)6-s + (0.539 − 2.59i)7-s + (−1.20 + 0.0728i)8-s + (2.30 − 1.87i)9-s + (0.318 − 3.95i)10-s + (−0.846 − 1.12i)11-s + (0.0857 + 0.213i)12-s + (3.27 + 6.24i)13-s + (0.890 − 4.75i)14-s + (0.360 − 0.0889i)15-s + (−4.86 + 0.393i)16-s + (−4.90 − 1.63i)17-s + ⋯ |
| L(s) = 1 | + (1.29 − 0.0260i)2-s + (0.0332 + 0.0932i)3-s + (0.669 − 0.0269i)4-s + (0.0974 − 0.964i)5-s + (0.0453 + 0.119i)6-s + (0.203 − 0.978i)7-s + (−0.425 + 0.0257i)8-s + (0.767 − 0.626i)9-s + (0.100 − 1.24i)10-s + (−0.255 − 0.339i)11-s + (0.0247 + 0.0614i)12-s + (0.908 + 1.73i)13-s + (0.237 − 1.27i)14-s + (0.0931 − 0.0229i)15-s + (−1.21 + 0.0982i)16-s + (−1.18 − 0.397i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.38809 - 0.917480i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.38809 - 0.917480i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.539 + 2.59i)T \) |
| 53 | \( 1 + (-6.06 - 4.02i)T \) |
| good | 2 | \( 1 + (-1.82 + 0.0367i)T + (1.99 - 0.0805i)T^{2} \) |
| 3 | \( 1 + (-0.0575 - 0.161i)T + (-2.32 + 1.89i)T^{2} \) |
| 5 | \( 1 + (-0.217 + 2.15i)T + (-4.89 - 1.00i)T^{2} \) |
| 11 | \( 1 + (0.846 + 1.12i)T + (-3.06 + 10.5i)T^{2} \) |
| 13 | \( 1 + (-3.27 - 6.24i)T + (-7.38 + 10.6i)T^{2} \) |
| 17 | \( 1 + (4.90 + 1.63i)T + (13.5 + 10.2i)T^{2} \) |
| 19 | \( 1 + (-5.31 - 5.76i)T + (-1.52 + 18.9i)T^{2} \) |
| 23 | \( 1 + (0.287 + 1.07i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.57 - 0.190i)T + (28.1 + 6.94i)T^{2} \) |
| 31 | \( 1 + (2.99 - 7.45i)T + (-22.3 - 21.4i)T^{2} \) |
| 37 | \( 1 + (0.140 + 1.73i)T + (-36.5 + 5.93i)T^{2} \) |
| 41 | \( 1 + (3.05 + 2.39i)T + (9.81 + 39.8i)T^{2} \) |
| 43 | \( 1 + (0.943 + 0.651i)T + (15.2 + 40.2i)T^{2} \) |
| 47 | \( 1 + (-0.428 + 2.63i)T + (-44.5 - 14.8i)T^{2} \) |
| 59 | \( 1 + (6.75 - 8.27i)T + (-11.8 - 57.8i)T^{2} \) |
| 61 | \( 1 + (3.23 + 1.61i)T + (36.6 + 48.7i)T^{2} \) |
| 67 | \( 1 + (0.208 + 0.192i)T + (5.39 + 66.7i)T^{2} \) |
| 71 | \( 1 + (1.17 + 6.41i)T + (-66.3 + 25.1i)T^{2} \) |
| 73 | \( 1 + (-9.96 + 4.97i)T + (43.8 - 58.3i)T^{2} \) |
| 79 | \( 1 + (9.01 - 4.96i)T + (42.2 - 66.7i)T^{2} \) |
| 83 | \( 1 + (4.51 - 4.51i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.07 - 0.423i)T + (81.8 - 34.8i)T^{2} \) |
| 97 | \( 1 + (-11.6 - 4.40i)T + (72.6 + 64.3i)T^{2} \) |
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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
−11.65247465537662188270901398903, −10.57903338451023200832201897162, −9.301534100464682646043262820318, −8.713511426090835535940137130018, −7.15571939126032482382471699523, −6.33801198808737007299473006729, −5.07100665132928872188147310895, −4.26957379452359462415677716068, −3.59584696032349169233286442474, −1.44562180702648094681835208779,
2.41092037665380463120370838677, 3.26016648095464292629548790939, 4.68232682962789412703374494798, 5.53810726353952266058311735742, 6.44869993256860346392888204189, 7.49060585994382776622866120365, 8.637466342131006613311775148655, 9.845209909142636432990391954024, 10.94013779501574187909615684578, 11.50572585443505479012923921615
