| L(s) = 1 | + (−1.19 + 0.752i)2-s + (−3.29 − 1.15i)3-s + (0.867 − 1.80i)4-s + (6.81 + 1.55i)5-s + (4.80 − 1.09i)6-s + (9.87 − 4.75i)7-s + (0.316 + 2.81i)8-s + (2.47 + 1.97i)9-s + (−9.33 + 3.26i)10-s + (0.486 − 4.31i)11-s + (−4.93 + 4.93i)12-s + (−1.82 + 1.45i)13-s + (−8.24 + 13.1i)14-s + (−20.6 − 12.9i)15-s + (−2.49 − 3.12i)16-s + (18.0 + 18.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.598 + 0.376i)2-s + (−1.09 − 0.383i)3-s + (0.216 − 0.450i)4-s + (1.36 + 0.311i)5-s + (0.801 − 0.182i)6-s + (1.41 − 0.679i)7-s + (0.0395 + 0.351i)8-s + (0.274 + 0.218i)9-s + (−0.933 + 0.326i)10-s + (0.0441 − 0.392i)11-s + (−0.410 + 0.410i)12-s + (−0.140 + 0.112i)13-s + (−0.588 + 0.937i)14-s + (−1.37 − 0.865i)15-s + (−0.155 − 0.195i)16-s + (1.06 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.859811 - 0.0661291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.859811 - 0.0661291i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.19 - 0.752i)T \) |
| 29 | \( 1 + (11.0 - 26.8i)T \) |
| good | 3 | \( 1 + (3.29 + 1.15i)T + (7.03 + 5.61i)T^{2} \) |
| 5 | \( 1 + (-6.81 - 1.55i)T + (22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 + (-9.87 + 4.75i)T + (30.5 - 38.3i)T^{2} \) |
| 11 | \( 1 + (-0.486 + 4.31i)T + (-117. - 26.9i)T^{2} \) |
| 13 | \( 1 + (1.82 - 1.45i)T + (37.6 - 164. i)T^{2} \) |
| 17 | \( 1 + (-18.0 - 18.0i)T + 289iT^{2} \) |
| 19 | \( 1 + (7.50 + 21.4i)T + (-282. + 225. i)T^{2} \) |
| 23 | \( 1 + (1.89 + 8.32i)T + (-476. + 229. i)T^{2} \) |
| 31 | \( 1 + (40.6 - 25.5i)T + (416. - 865. i)T^{2} \) |
| 37 | \( 1 + (6.33 + 56.1i)T + (-1.33e3 + 304. i)T^{2} \) |
| 41 | \( 1 + (-14.8 + 14.8i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (31.0 - 49.3i)T + (-802. - 1.66e3i)T^{2} \) |
| 47 | \( 1 + (-23.6 - 2.66i)T + (2.15e3 + 491. i)T^{2} \) |
| 53 | \( 1 + (16.2 - 71.3i)T + (-2.53e3 - 1.21e3i)T^{2} \) |
| 59 | \( 1 + 59.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + (83.9 + 29.3i)T + (2.90e3 + 2.32e3i)T^{2} \) |
| 67 | \( 1 + (-16.1 - 12.9i)T + (998. + 4.37e3i)T^{2} \) |
| 71 | \( 1 + (49.1 - 39.2i)T + (1.12e3 - 4.91e3i)T^{2} \) |
| 73 | \( 1 + (-43.1 - 27.1i)T + (2.31e3 + 4.80e3i)T^{2} \) |
| 79 | \( 1 + (-121. + 13.7i)T + (6.08e3 - 1.38e3i)T^{2} \) |
| 83 | \( 1 + (23.5 + 11.3i)T + (4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (23.3 - 14.6i)T + (3.43e3 - 7.13e3i)T^{2} \) |
| 97 | \( 1 + (-63.9 + 22.3i)T + (7.35e3 - 5.86e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.69627961315613667590583394177, −14.05558384411424327541154911091, −12.58293412968281906992540829057, −10.98422219336421502214028745098, −10.63804498299518428399301540355, −9.001699926660065897436204623014, −7.42923630585238285107535530614, −6.18396374419361282729778528277, −5.20301239254687063912074255280, −1.48419918015347991423878223377,
1.85293685650541723875713156578, 5.02612831923483375426242783284, 5.86673036927100664187085659617, 7.933251246387934122282678160801, 9.417789023167773158310936631190, 10.30787453595133475094359704284, 11.48891480455787522980583127499, 12.23909879299819816586511595999, 13.80984858894711076331035203005, 15.02763966362198210493100957199
