| L(s) = 1 | + (0.732 − 2.73i)2-s + (1.03 + 0.600i)3-s + (−6.92 − 4i)4-s + (1.65 + 6.17i)5-s + (2.40 − 2.40i)6-s + (28.4 − 49.2i)7-s + (−16 + 15.9i)8-s + (−39.7 − 68.8i)9-s + 18.0·10-s − 60.5i·11-s + (−4.80 − 8.31i)12-s + (−37.8 − 141. i)13-s + (−113. − 113. i)14-s + (−1.98 + 7.41i)15-s + (31.9 + 55.4i)16-s + (−151. − 40.7i)17-s + ⋯ |
| L(s) = 1 | + (0.183 − 0.683i)2-s + (0.115 + 0.0666i)3-s + (−0.433 − 0.250i)4-s + (0.0661 + 0.246i)5-s + (0.0666 − 0.0666i)6-s + (0.579 − 1.00i)7-s + (−0.250 + 0.249i)8-s + (−0.491 − 0.850i)9-s + 0.180·10-s − 0.500i·11-s + (−0.0333 − 0.0577i)12-s + (−0.224 − 0.836i)13-s + (−0.579 − 0.579i)14-s + (−0.00882 + 0.0329i)15-s + (0.124 + 0.216i)16-s + (−0.525 − 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.735153 - 1.40561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.735153 - 1.40561i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.732 + 2.73i)T \) |
| 37 | \( 1 + (-1.27e3 - 504. i)T \) |
| good | 3 | \( 1 + (-1.03 - 0.600i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-1.65 - 6.17i)T + (-541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (-28.4 + 49.2i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 60.5iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (37.8 + 141. i)T + (-2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (151. + 40.7i)T + (7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-31.0 - 116. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (59.6 - 59.6i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (266. + 266. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-154. - 154. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (-1.46e3 - 844. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (263. - 263. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 2.74e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-2.61e3 - 4.52e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (3.56e3 + 956. i)T + (1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (2.42e3 - 649. i)T + (1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (2.28e3 + 1.31e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (174. - 302. i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + 4.75e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (1.51e3 + 5.66e3i)T + (-3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-2.16e3 - 3.75e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.11e3 + 4.15e3i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-7.08e3 + 7.08e3i)T - 8.85e7iT^{2} \) |
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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
−13.51466923640496428216451154560, −12.27450330660973119072385818744, −11.14090805172001524452058071684, −10.34317544023453074109945289473, −9.024100083598179218173072085191, −7.69535033654551796973123254410, −6.04103001267184160570080966696, −4.36589372049180415988393355944, −2.97016019302283859460101072972, −0.792499207295587763313356164869,
2.26586869600431764072523519522, 4.59617856377615027527995074525, 5.64349933192520127891035032548, 7.18855475408555504091173230809, 8.449306734546260892244729711500, 9.268504457378202374913622478113, 11.03086026639705966318287751401, 12.15540626148314918849504042477, 13.28161738666379867027303054565, 14.37115784377423682126387656435
