| L(s) = 1 | + (−2.28 + 6.28i)3-s + (−2.03 − 11.5i)5-s + (38.6 − 66.9i)7-s + (27.7 + 23.3i)9-s + (79.3 + 137. i)11-s + (65.1 + 179. i)13-s + (77.0 + 13.5i)15-s + (343. − 288. i)17-s + (−191. + 306. i)19-s + (332. + 395. i)21-s + (99.9 − 567. i)23-s + (458. − 167. i)25-s + (−679. + 392. i)27-s + (447. − 533. i)29-s + (533. + 308. i)31-s + ⋯ |
| L(s) = 1 | + (−0.254 + 0.698i)3-s + (−0.0812 − 0.460i)5-s + (0.788 − 1.36i)7-s + (0.343 + 0.287i)9-s + (0.655 + 1.13i)11-s + (0.385 + 1.05i)13-s + (0.342 + 0.0603i)15-s + (1.18 − 0.997i)17-s + (−0.530 + 0.847i)19-s + (0.753 + 0.897i)21-s + (0.189 − 1.07i)23-s + (0.734 − 0.267i)25-s + (−0.931 + 0.537i)27-s + (0.531 − 0.633i)29-s + (0.555 + 0.320i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.72656 + 0.272352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72656 + 0.272352i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (191. - 306. i)T \) |
| good | 3 | \( 1 + (2.28 - 6.28i)T + (-62.0 - 52.0i)T^{2} \) |
| 5 | \( 1 + (2.03 + 11.5i)T + (-587. + 213. i)T^{2} \) |
| 7 | \( 1 + (-38.6 + 66.9i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-79.3 - 137. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-65.1 - 179. i)T + (-2.18e4 + 1.83e4i)T^{2} \) |
| 17 | \( 1 + (-343. + 288. i)T + (1.45e4 - 8.22e4i)T^{2} \) |
| 23 | \( 1 + (-99.9 + 567. i)T + (-2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-447. + 533. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-533. - 308. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 2.28e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (792. - 2.17e3i)T + (-2.16e6 - 1.81e6i)T^{2} \) |
| 43 | \( 1 + (-465. - 2.64e3i)T + (-3.21e6 + 1.16e6i)T^{2} \) |
| 47 | \( 1 + (1.39e3 + 1.17e3i)T + (8.47e5 + 4.80e6i)T^{2} \) |
| 53 | \( 1 + (1.88e3 + 333. i)T + (7.41e6 + 2.69e6i)T^{2} \) |
| 59 | \( 1 + (1.32e3 + 1.57e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-228. + 1.29e3i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (3.21e3 - 3.83e3i)T + (-3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (4.66e3 - 822. i)T + (2.38e7 - 8.69e6i)T^{2} \) |
| 73 | \( 1 + (-6.55e3 - 2.38e3i)T + (2.17e7 + 1.82e7i)T^{2} \) |
| 79 | \( 1 + (2.01e3 - 5.54e3i)T + (-2.98e7 - 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-2.41e3 + 4.18e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.67e3 + 4.59e3i)T + (-4.80e7 + 4.03e7i)T^{2} \) |
| 97 | \( 1 + (-42.3 - 50.4i)T + (-1.53e7 + 8.71e7i)T^{2} \) |
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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
−14.07904025366318278104830376531, −12.65907962776248098638691979536, −11.49989613424743086644683866535, −10.40624341883946621441936732158, −9.569213178620163461409398253910, −7.989558113072451761923058130935, −6.82637350980586579714073286093, −4.72643604028979589550983472784, −4.21260213949640509162219283871, −1.34868005748958072092044441039,
1.29532781544470612463841250095, 3.22499536306591290953231582320, 5.47046453987085750140404943025, 6.44492105796614185342866769837, 7.964282004606106910010278131303, 8.916183589874714459625888955376, 10.61484290731938919335544776312, 11.69207363949535112484114724275, 12.45269088375045711422485903202, 13.63764586348847021901129766487
