| L(s) = 1 | + (1.50 − 8.51i)3-s + (−8.90 − 7.46i)5-s + (70.8 + 122. i)7-s + (158. + 57.5i)9-s + (194. − 337. i)11-s + (−114. − 650. i)13-s + (−76.9 + 64.5i)15-s + (368. − 134. i)17-s + (1.14e3 − 1.07e3i)19-s + (1.15e3 − 419. i)21-s + (−396. + 332. i)23-s + (−519. − 2.94e3i)25-s + (1.77e3 − 3.07e3i)27-s + (6.19e3 + 2.25e3i)29-s + (−306. − 531. i)31-s + ⋯ |
| L(s) = 1 | + (0.0963 − 0.546i)3-s + (−0.159 − 0.133i)5-s + (0.546 + 0.947i)7-s + (0.650 + 0.236i)9-s + (0.485 − 0.840i)11-s + (−0.188 − 1.06i)13-s + (−0.0882 + 0.0740i)15-s + (0.309 − 0.112i)17-s + (0.729 − 0.684i)19-s + (0.570 − 0.207i)21-s + (−0.156 + 0.131i)23-s + (−0.166 − 0.942i)25-s + (0.469 − 0.812i)27-s + (1.36 + 0.497i)29-s + (−0.0573 − 0.0992i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.85857 - 0.761820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85857 - 0.761820i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.14e3 + 1.07e3i)T \) |
| good | 3 | \( 1 + (-1.50 + 8.51i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (8.90 + 7.46i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-70.8 - 122. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-194. + 337. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (114. + 650. i)T + (-3.48e5 + 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-368. + 134. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 23 | \( 1 + (396. - 332. i)T + (1.11e6 - 6.33e6i)T^{2} \) |
| 29 | \( 1 + (-6.19e3 - 2.25e3i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (306. + 531. i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 9.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (2.68e3 - 1.52e4i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-779. - 653. i)T + (2.55e7 + 1.44e8i)T^{2} \) |
| 47 | \( 1 + (1.59e4 + 5.79e3i)T + (1.75e8 + 1.47e8i)T^{2} \) |
| 53 | \( 1 + (2.03e4 - 1.70e4i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-1.88e4 + 6.85e3i)T + (5.47e8 - 4.59e8i)T^{2} \) |
| 61 | \( 1 + (3.86e4 - 3.24e4i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.38e4 - 1.23e4i)T + (1.03e9 + 8.67e8i)T^{2} \) |
| 71 | \( 1 + (1.90e4 + 1.60e4i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 + (5.13e3 - 2.91e4i)T + (-1.94e9 - 7.09e8i)T^{2} \) |
| 79 | \( 1 + (-5.03e3 + 2.85e4i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (3.68e4 + 6.38e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-4.80e3 - 2.72e4i)T + (-5.24e9 + 1.90e9i)T^{2} \) |
| 97 | \( 1 + (-1.36e4 + 4.95e3i)T + (6.57e9 - 5.51e9i)T^{2} \) |
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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
−13.32062914752934565219672634861, −12.29918955663097890131187514576, −11.42368772625700104787947158902, −10.00441902785854713633553913125, −8.582514870940924326704069531613, −7.72213448821133368260017061051, −6.20540606941943512489295983950, −4.84264593293299898989360244463, −2.82338353860892561935026642072, −1.05924077143815168098117818958,
1.45714729122274510819235583417, 3.80956943196790076249666160877, 4.71446035884795267216601889442, 6.74770895539102314804154969560, 7.74385750493751958019286273558, 9.409069420929278619417848852481, 10.17858282043146201386995590880, 11.40807637256949361800727384345, 12.47594685403304301348613455823, 13.93462957325510873292145339922
