L(s) = 1 | + (−13.1 − 7.59i)3-s + (19.9 − 34.5i)5-s − 53.3·7-s + (74.8 + 129. i)9-s + 9.01·11-s + (−5.39 + 3.11i)13-s + (−524. + 302. i)15-s + (−261. + 453. i)17-s + (−329. − 147. i)19-s + (701. + 405. i)21-s + (204. + 353. i)23-s + (−481. − 834. i)25-s − 1.04e3i·27-s + (1.17e3 − 679. i)29-s + 214. i·31-s + ⋯ |
L(s) = 1 | + (−1.46 − 0.843i)3-s + (0.797 − 1.38i)5-s − 1.08·7-s + (0.923 + 1.59i)9-s + 0.0744·11-s + (−0.0319 + 0.0184i)13-s + (−2.32 + 1.34i)15-s + (−0.905 + 1.56i)17-s + (−0.913 − 0.407i)19-s + (1.59 + 0.918i)21-s + (0.385 + 0.668i)23-s + (−0.770 − 1.33i)25-s − 1.42i·27-s + (1.39 − 0.807i)29-s + 0.223i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0854082 + 0.189773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0854082 + 0.189773i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (329. + 147. i)T \) |
good | 3 | \( 1 + (13.1 + 7.59i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-19.9 + 34.5i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + 53.3T + 2.40e3T^{2} \) |
| 11 | \( 1 - 9.01T + 1.46e4T^{2} \) |
| 13 | \( 1 + (5.39 - 3.11i)T + (1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + (261. - 453. i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 23 | \( 1 + (-204. - 353. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-1.17e3 + 679. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 - 214. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.22e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (666. + 384. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.14e3 - 1.98e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (1.74e3 + 3.01e3i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (4.81e3 - 2.78e3i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (2.13e3 + 1.23e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.47e3 + 2.54e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-1.47e3 + 851. i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (2.98e3 + 1.72e3i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (4.12e3 - 7.13e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-7.35e3 - 4.24e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + 2.75e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-5.98e3 + 3.45e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (1.05e4 + 6.07e3i)T + (4.42e7 + 7.66e7i)T^{2} \) |
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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
−12.85741818362960343520632632943, −12.36885184769041962464781841927, −10.98576962728199557312573536235, −9.777862292482920408860725444361, −8.471521467673060574588886942938, −6.59499199767520350070577559274, −5.97147813575086799187138999123, −4.68120545399771299612461343940, −1.62054595554836339707053918007, −0.12515793293169903243008735216,
2.99930136862071314558739584448, 4.83136843203201552588995301267, 6.39939945224618811308929567673, 6.64049174950814692596212586729, 9.390111498669414888700672989041, 10.26079823910724971047865384358, 10.88237294825744514759415633169, 12.00221141985894554390007342674, 13.33899835154470546082960188114, 14.59689858427891823211034892181