| 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 | \( 1 + (-63.9 - 22.3i)T + (7.35e3 + 5.86e3i)T^{2} \) |
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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
−15.02763966362198210493100957199, −13.80984858894711076331035203005, −12.23909879299819816586511595999, −11.48891480455787522980583127499, −10.30787453595133475094359704284, −9.417789023167773158310936631190, −7.933251246387934122282678160801, −5.86673036927100664187085659617, −5.02612831923483375426242783284, −1.85293685650541723875713156578,
1.48419918015347991423878223377, 5.20301239254687063912074255280, 6.18396374419361282729778528277, 7.42923630585238285107535530614, 9.001699926660065897436204623014, 10.63804498299518428399301540355, 10.98422219336421502214028745098, 12.58293412968281906992540829057, 14.05558384411424327541154911091, 14.69627961315613667590583394177
