| L(s) = 1 | + (−0.579 + 2.66i)2-s + (−2.04 + 1.52i)3-s + (−4.93 − 2.25i)4-s + (2.12 + 0.709i)5-s + (−2.89 − 6.32i)6-s + (−0.134 + 1.87i)7-s + (5.60 − 7.48i)8-s + (0.990 − 3.37i)9-s + (−3.11 + 5.23i)10-s + (−0.338 + 0.526i)11-s + (13.5 − 2.94i)12-s + (−2.62 + 0.187i)13-s + (−4.92 − 1.44i)14-s + (−5.41 + 1.79i)15-s + (9.57 + 11.0i)16-s + (0.245 − 0.657i)17-s + ⋯ |
| L(s) = 1 | + (−0.409 + 1.88i)2-s + (−1.17 + 0.883i)3-s + (−2.46 − 1.12i)4-s + (0.948 + 0.317i)5-s + (−1.17 − 2.58i)6-s + (−0.0507 + 0.709i)7-s + (1.98 − 2.64i)8-s + (0.330 − 1.12i)9-s + (−0.985 + 1.65i)10-s + (−0.102 + 0.158i)11-s + (3.90 − 0.850i)12-s + (−0.727 + 0.0520i)13-s + (−1.31 − 0.386i)14-s + (−1.39 + 0.463i)15-s + (2.39 + 2.76i)16-s + (0.0595 − 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.238440 - 0.441954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.238440 - 0.441954i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.12 - 0.709i)T \) |
| 23 | \( 1 + (3.33 - 3.44i)T \) |
| good | 2 | \( 1 + (0.579 - 2.66i)T + (-1.81 - 0.830i)T^{2} \) |
| 3 | \( 1 + (2.04 - 1.52i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (0.134 - 1.87i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (0.338 - 0.526i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (2.62 - 0.187i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-0.245 + 0.657i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (0.374 - 0.819i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.285 - 0.130i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.816 - 5.68i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.127 - 0.233i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-6.57 + 1.92i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-6.98 - 9.32i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-3.96 + 3.96i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.60 - 0.472i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (2.23 + 1.93i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (9.02 - 1.29i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (13.7 + 2.99i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-5.95 + 3.82i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (6.69 - 2.49i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-6.21 + 7.16i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (0.0389 - 0.0212i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (0.678 - 4.72i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-9.14 - 4.99i)T + (52.4 + 81.6i)T^{2} \) |
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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
−14.65745264394187198687032646583, −13.70154623428484176025425458207, −12.30244978617833379347798410960, −10.61010733143732331733347908285, −9.770442921644334363667495895585, −9.044652026886008630603296683745, −7.43449902408080320240271637166, −6.15373520705704231085976440578, −5.60118328484108183443265467650, −4.64052954795198168329691996913,
0.75913600923154005491167631284, 2.26968813677609194912601584614, 4.43377062670301981073560843590, 5.81082135733700591368015361680, 7.53622107096943701102983558551, 9.051574769954332686871080584979, 10.19235413413752190475205913328, 10.80167234644681439949626079854, 11.91374420698017730388777543247, 12.59200559494016912784518518104
