| L(s) = 1 | + (6.23 − 5.23i)3-s + (−51.5 − 18.7i)5-s + (−28.9 + 50.2i)7-s + (−30.6 + 174. i)9-s + (274. + 474. i)11-s + (−42.6 − 35.8i)13-s + (−419. + 152. i)15-s + (38.5 + 218. i)17-s + (1.00e3 + 1.20e3i)19-s + (81.9 + 464. i)21-s + (−1.36e3 + 498. i)23-s + (−86.4 − 72.5i)25-s + (1.70e3 + 2.95e3i)27-s + (−879. + 4.98e3i)29-s + (−2.84e3 + 4.93e3i)31-s + ⋯ |
| L(s) = 1 | + (0.399 − 0.335i)3-s + (−0.922 − 0.335i)5-s + (−0.223 + 0.387i)7-s + (−0.126 + 0.716i)9-s + (0.682 + 1.18i)11-s + (−0.0700 − 0.0587i)13-s + (−0.481 + 0.175i)15-s + (0.0323 + 0.183i)17-s + (0.640 + 0.767i)19-s + (0.0405 + 0.229i)21-s + (−0.539 + 0.196i)23-s + (−0.0276 − 0.0232i)25-s + (0.450 + 0.780i)27-s + (−0.194 + 1.10i)29-s + (−0.532 + 0.922i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0131 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0131 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.827949 + 0.838930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.827949 + 0.838930i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.00e3 - 1.20e3i)T \) |
| good | 3 | \( 1 + (-6.23 + 5.23i)T + (42.1 - 239. i)T^{2} \) |
| 5 | \( 1 + (51.5 + 18.7i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (28.9 - 50.2i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-274. - 474. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (42.6 + 35.8i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-38.5 - 218. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 23 | \( 1 + (1.36e3 - 498. i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (879. - 4.98e3i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (2.84e3 - 4.93e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 7.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-6.03e3 + 5.06e3i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (1.30e4 + 4.76e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-3.72e3 + 2.11e4i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 + (-7.95e3 + 2.89e3i)T + (3.20e8 - 2.68e8i)T^{2} \) |
| 59 | \( 1 + (1.78e3 + 1.01e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (1.79e4 - 6.52e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-7.69e3 + 4.36e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-6.44e3 - 2.34e3i)T + (1.38e9 + 1.15e9i)T^{2} \) |
| 73 | \( 1 + (1.61e4 - 1.35e4i)T + (3.59e8 - 2.04e9i)T^{2} \) |
| 79 | \( 1 + (-4.22e4 + 3.54e4i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (3.55e4 - 6.15e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-9.83e4 - 8.25e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-3.16e4 - 1.79e5i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−13.84907651154111521136539319867, −12.47478443780360302269682357084, −11.97228114219352428618883710019, −10.45919417282811977898936546957, −9.088616783830609081581871868853, −7.993388889561208415969442634080, −7.02869569116612757278591698706, −5.16641932543060430317779310122, −3.66802954480275815545412893564, −1.80364517492837631557580273788,
0.49231349996982108535282456309, 3.18428234562195839136746458301, 4.09596611354473731977429516132, 6.14428763683216952033350448670, 7.46318649485629570094232288364, 8.706405919373506071408408252695, 9.760041766868447001555911359148, 11.23418640238376867736287289376, 11.88314934866021768708294441825, 13.44990062334293628703795248514
