| L(s) = 1 | + (3.33 − 1.08i)2-s + (−3.80 − 5.23i)3-s + (3.48 − 2.53i)4-s + (10.7 + 3.21i)5-s + (−18.3 − 13.3i)6-s + 25.7i·7-s + (−7.61 + 10.4i)8-s + (−4.60 + 14.1i)9-s + (39.2 − 0.874i)10-s + (−17.5 − 53.8i)11-s + (−26.5 − 8.61i)12-s + (0.673 + 0.218i)13-s + (27.9 + 85.9i)14-s + (−23.8 − 68.3i)15-s + (−24.6 + 75.9i)16-s + (−10.2 + 14.1i)17-s + ⋯ |
| L(s) = 1 | + (1.17 − 0.383i)2-s + (−0.732 − 1.00i)3-s + (0.435 − 0.316i)4-s + (0.957 + 0.287i)5-s + (−1.25 − 0.908i)6-s + 1.39i·7-s + (−0.336 + 0.463i)8-s + (−0.170 + 0.525i)9-s + (1.24 − 0.0276i)10-s + (−0.479 − 1.47i)11-s + (−0.638 − 0.207i)12-s + (0.0143 + 0.00467i)13-s + (0.532 + 1.64i)14-s + (−0.411 − 1.17i)15-s + (−0.385 + 1.18i)16-s + (−0.146 + 0.201i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.52106 - 0.674745i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52106 - 0.674745i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-10.7 - 3.21i)T \) |
| good | 2 | \( 1 + (-3.33 + 1.08i)T + (6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (3.80 + 5.23i)T + (-8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 - 25.7iT - 343T^{2} \) |
| 11 | \( 1 + (17.5 + 53.8i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-0.673 - 0.218i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (10.2 - 14.1i)T + (-1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (39.7 + 28.8i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-14.1 + 4.59i)T + (9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-183. + 133. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (82.5 + 59.9i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (4.35 + 1.41i)T + (4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (50.1 - 154. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 522. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-358. - 493. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-82.3 - 113. i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (25.8 - 79.6i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-82.0 - 252. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (45.3 - 62.4i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-586. + 426. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (304. - 98.8i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (39.6 - 28.8i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (433. - 596. i)T + (-1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (372. + 1.14e3i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (392. + 539. i)T + (-2.82e5 + 8.68e5i)T^{2} \) |
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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
−17.22046904935058047775433833676, −15.36879043284847858469716720250, −13.93251914315296574191758551585, −13.06524025854089324964184612371, −12.11988511703349085154717067879, −11.03827427008753862164819221096, −8.731756405375358247518259962824, −6.19950291842615722071601988357, −5.56219949195839414247538238336, −2.55361191311995916873280032239,
4.31366445498947956753894519635, 5.20954204005562255002647395395, 6.86049832244901097663868467222, 9.766987325307827049758566621676, 10.54773257516372038336837961407, 12.53758456639009406725297904569, 13.59687763166068099679026997914, 14.70526687817731724866526049414, 15.96904358320611697015361337127, 16.94012078584795379924626896533
