L(s) = 1 | + (1.07 − 1.86i)3-s + (−10.6 + 18.4i)5-s − 18.8·7-s + (11.1 + 19.3i)9-s + 8.41·11-s + (12.3 + 21.4i)13-s + (22.8 + 39.6i)15-s + (−18.3 + 31.7i)17-s + (−76.9 − 30.5i)19-s + (−20.2 + 35.1i)21-s + (19.6 + 34.0i)23-s + (−163. − 283. i)25-s + 106.·27-s + (57.8 + 100. i)29-s + 304.·31-s + ⋯ |
L(s) = 1 | + (0.207 − 0.358i)3-s + (−0.950 + 1.64i)5-s − 1.01·7-s + (0.414 + 0.717i)9-s + 0.230·11-s + (0.264 + 0.457i)13-s + (0.394 + 0.682i)15-s + (−0.261 + 0.452i)17-s + (−0.929 − 0.369i)19-s + (−0.210 + 0.365i)21-s + (0.178 + 0.308i)23-s + (−1.30 − 2.26i)25-s + 0.757·27-s + (0.370 + 0.642i)29-s + 1.76·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.547128 + 0.789118i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.547128 + 0.789118i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (76.9 + 30.5i)T \) |
good | 3 | \( 1 + (-1.07 + 1.86i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (10.6 - 18.4i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + 18.8T + 343T^{2} \) |
| 11 | \( 1 - 8.41T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-12.3 - 21.4i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (18.3 - 31.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 23 | \( 1 + (-19.6 - 34.0i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-57.8 - 100. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 304.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 265.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-172. + 299. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (108. - 187. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (47.7 + 82.7i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-347. - 601. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-221. + 384. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-188. - 327. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-412. - 714. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (207. - 358. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (66.0 - 114. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (349. - 605. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (56.8 + 98.4i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-207. + 359. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.33216931297269536902640554715, −13.36556624486636174054654293290, −12.13444457423147371298278298199, −10.93983224187604703637934372570, −10.16684229748022747074277830739, −8.428801840290306283225260015380, −7.10469176615125807101278388557, −6.51463926121772038012649632073, −4.01331125702110887204442097023, −2.66253091320932121468141475933,
0.60271819212407974061567620078, 3.62659562078917518057802276053, 4.71190935658767693489432560943, 6.48807538841823306681311919071, 8.185026555979857801859130782558, 9.049932466560510957857855986968, 10.06047852645765847022157514593, 11.81719331465351958564687072550, 12.56770975420196548062436382120, 13.36894756261868775712816359189