| L(s) = 1 | + (−3.78 − 2.18i)2-s + (2.97 − 4.26i)3-s + (5.54 + 9.60i)4-s + (7.71 + 2.06i)5-s + (−20.5 + 9.62i)6-s + (−3.71 + 0.996i)7-s − 13.5i·8-s + (−9.31 − 25.3i)9-s + (−24.6 − 24.6i)10-s + (37.8 − 10.1i)11-s + (57.4 + 4.93i)12-s + (−19.2 − 33.3i)13-s + (16.2 + 4.35i)14-s + (31.7 − 26.7i)15-s + (14.8 − 25.7i)16-s + (67.1 − 20.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 − 0.772i)2-s + (0.572 − 0.820i)3-s + (0.693 + 1.20i)4-s + (0.690 + 0.184i)5-s + (−1.39 + 0.654i)6-s + (−0.200 + 0.0537i)7-s − 0.596i·8-s + (−0.344 − 0.938i)9-s + (−0.780 − 0.780i)10-s + (1.03 − 0.277i)11-s + (1.38 + 0.118i)12-s + (−0.411 − 0.712i)13-s + (0.310 + 0.0830i)14-s + (0.546 − 0.460i)15-s + (0.232 − 0.402i)16-s + (0.958 − 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.313585 - 1.00160i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.313585 - 1.00160i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.97 + 4.26i)T \) |
| 17 | \( 1 + (-67.1 + 20.0i)T \) |
| good | 2 | \( 1 + (3.78 + 2.18i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-7.71 - 2.06i)T + (108. + 62.5i)T^{2} \) |
| 7 | \( 1 + (3.71 - 0.996i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-37.8 + 10.1i)T + (1.15e3 - 665.5i)T^{2} \) |
| 13 | \( 1 + (19.2 + 33.3i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 19 | \( 1 + 36.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-7.55 + 28.1i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (-4.91 - 18.3i)T + (-2.11e4 + 1.21e4i)T^{2} \) |
| 31 | \( 1 + (187. + 50.3i)T + (2.57e4 + 1.48e4i)T^{2} \) |
| 37 | \( 1 + (-80.7 + 80.7i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (-65.4 + 244. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-88.3 - 51.0i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (150. - 260. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 312. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-605. + 349. i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-301. + 80.9i)T + (1.96e5 - 1.13e5i)T^{2} \) |
| 67 | \( 1 + (162. + 281. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-198. + 198. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (156. - 156. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (-797. + 213. i)T + (4.26e5 - 2.46e5i)T^{2} \) |
| 83 | \( 1 + (878. + 506. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 697.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-444. - 1.65e3i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−12.00726550420596782169794148273, −10.96655569821603589127064230597, −9.735599252355466093751024978514, −9.235432389914607491254405965195, −8.138432996586253559401773282664, −7.17512770082075169263307006610, −5.85852756316607790670027399260, −3.25295241786035724177886120065, −2.07881917034960381011917339775, −0.76202082879112899348551385072,
1.68782752361131726106947201635, 3.82351000982752948654574565191, 5.52609180649942751753971500109, 6.78296162781145246518842403070, 7.938417813981334807291324014855, 8.985679789567030692154716922468, 9.654799076053152494841126987517, 10.16189430567221840891871973997, 11.57887017845900600713394126570, 13.10666269897626465635500849126
