| L(s) = 1 | + (1.19 + 0.752i)2-s + (−5.22 + 1.82i)3-s + (0.867 + 1.80i)4-s + (−6.53 + 1.49i)5-s + (−7.62 − 1.74i)6-s + (4.88 + 2.35i)7-s + (−0.316 + 2.81i)8-s + (16.8 − 13.4i)9-s + (−8.95 − 3.13i)10-s + (2.09 + 18.5i)11-s + (−7.82 − 7.82i)12-s + (−4.48 − 3.57i)13-s + (4.08 + 6.49i)14-s + (31.4 − 19.7i)15-s + (−2.49 + 3.12i)16-s + (−4.23 + 4.23i)17-s + ⋯ |
| L(s) = 1 | + (0.598 + 0.376i)2-s + (−1.74 + 0.608i)3-s + (0.216 + 0.450i)4-s + (−1.30 + 0.298i)5-s + (−1.27 − 0.290i)6-s + (0.698 + 0.336i)7-s + (−0.0395 + 0.351i)8-s + (1.87 − 1.49i)9-s + (−0.895 − 0.313i)10-s + (0.190 + 1.69i)11-s + (−0.651 − 0.651i)12-s + (−0.345 − 0.275i)13-s + (0.291 + 0.464i)14-s + (2.09 − 1.31i)15-s + (−0.155 + 0.195i)16-s + (−0.249 + 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.218379 + 0.672179i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.218379 + 0.672179i\) |
| \(L(2)\) |
|
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 | \( 1 + (5.22 - 1.82i)T + (7.03 - 5.61i)T^{2} \) |
| 5 | \( 1 + (6.53 - 1.49i)T + (22.5 - 10.8i)T^{2} \) |
| 7 | \( 1 + (-4.88 - 2.35i)T + (30.5 + 38.3i)T^{2} \) |
| 11 | \( 1 + (-2.09 - 18.5i)T + (-117. + 26.9i)T^{2} \) |
| 13 | \( 1 + (4.48 + 3.57i)T + (37.6 + 164. i)T^{2} \) |
| 17 | \( 1 + (4.23 - 4.23i)T - 289iT^{2} \) |
| 19 | \( 1 + (-2.52 + 7.20i)T + (-282. - 225. i)T^{2} \) |
| 23 | \( 1 + (2.93 - 12.8i)T + (-476. - 229. i)T^{2} \) |
| 31 | \( 1 + (-27.2 - 17.1i)T + (416. + 865. i)T^{2} \) |
| 37 | \( 1 + (0.341 - 3.03i)T + (-1.33e3 - 304. i)T^{2} \) |
| 41 | \( 1 + (-39.5 - 39.5i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-9.93 - 15.8i)T + (-802. + 1.66e3i)T^{2} \) |
| 47 | \( 1 + (58.1 - 6.54i)T + (2.15e3 - 491. i)T^{2} \) |
| 53 | \( 1 + (-8.43 - 36.9i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 - 69.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-77.1 + 26.9i)T + (2.90e3 - 2.32e3i)T^{2} \) |
| 67 | \( 1 + (-24.4 + 19.5i)T + (998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (64.4 + 51.4i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (-23.4 + 14.7i)T + (2.31e3 - 4.80e3i)T^{2} \) |
| 79 | \( 1 + (-15.1 - 1.70i)T + (6.08e3 + 1.38e3i)T^{2} \) |
| 83 | \( 1 + (-13.9 + 6.69i)T + (4.29e3 - 5.38e3i)T^{2} \) |
| 89 | \( 1 + (-27.4 - 17.2i)T + (3.43e3 + 7.13e3i)T^{2} \) |
| 97 | \( 1 + (35.3 + 12.3i)T + (7.35e3 + 5.86e3i)T^{2} \) |
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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
−15.36714137023980244825753396038, −14.89710156673197511262959094454, −12.65877326567850711498220235211, −11.83104189156934654019974091271, −11.30052487232254640482101776986, −9.938865891815664557940442846045, −7.72273846463769113583641962280, −6.59948521763606701132911175372, −4.99597774836789984898030582382, −4.25093837881432373250802915739,
0.72320804983555171771390698381, 4.20994267511394357788250853986, 5.43132348701978683065158261873, 6.79987699158228759801031603484, 8.141250198724736485852947017092, 10.63846684269649568343529509486, 11.46255891318045837369282156665, 11.87690502441572119399934514297, 12.99874507558032793796730915060, 14.27741015801855865647155475966
