| L(s) = 1 | + (−0.891 − 1.48i)2-s + (−0.506 + 2.95i)3-s + (0.472 − 0.892i)4-s + (1.05 + 9.65i)5-s + (4.83 − 1.88i)6-s + (−1.35 + 0.456i)7-s + (−8.65 + 0.469i)8-s + (−8.48 − 2.99i)9-s + (13.3 − 10.1i)10-s + (−12.7 − 8.64i)11-s + (2.39 + 1.85i)12-s + (−9.09 − 8.61i)13-s + (1.88 + 1.59i)14-s + (−29.0 − 1.78i)15-s + (6.14 + 9.05i)16-s + (−1.14 + 3.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.445 − 0.740i)2-s + (−0.168 + 0.985i)3-s + (0.118 − 0.223i)4-s + (0.210 + 1.93i)5-s + (0.805 − 0.314i)6-s + (−0.193 + 0.0651i)7-s + (−1.08 + 0.0586i)8-s + (−0.942 − 0.332i)9-s + (1.33 − 1.01i)10-s + (−1.15 − 0.785i)11-s + (0.199 + 0.154i)12-s + (−0.699 − 0.662i)13-s + (0.134 + 0.114i)14-s + (−1.93 − 0.119i)15-s + (0.383 + 0.565i)16-s + (−0.0673 + 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.193812 + 0.486479i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.193812 + 0.486479i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.506 - 2.95i)T \) |
| 59 | \( 1 + (56.9 + 15.2i)T \) |
| good | 2 | \( 1 + (0.891 + 1.48i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (-1.05 - 9.65i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (1.35 - 0.456i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (12.7 + 8.64i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (9.09 + 8.61i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (1.14 - 3.39i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (-3.10 - 18.9i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (-24.8 + 6.90i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (17.7 - 29.5i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (1.76 - 10.7i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-0.114 + 2.10i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (-44.0 - 12.2i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (-34.3 - 50.7i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (1.22 - 11.2i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-14.0 + 18.4i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (57.9 - 34.8i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (-1.18 - 21.9i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-4.68 + 43.1i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (-76.5 + 90.1i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (22.9 - 57.5i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (-40.3 - 87.2i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (47.3 - 78.6i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (74.8 + 88.1i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−12.53118409503196588482388408968, −11.06447708630537930211165697446, −10.88317735035419650727434788201, −10.17494508961386253689525406643, −9.365070837791699028866369076989, −7.79285300207448365684349142619, −6.32219513010447818681353370520, −5.47868838008271366706551940871, −3.24731050854147274168657162549, −2.72658460717567960846218763351,
0.33916413983883867074826899240, 2.28649605304980599718267617073, 4.78452158347355392798553382418, 5.74234991298160768031104616897, 7.17202353193151549308781122335, 7.80026749950756730889926035889, 8.867004576096241074707411147490, 9.512415886831713999385633970743, 11.46615640939789889572001923957, 12.40506108542324694350349197853
