| L(s) = 1 | + (−2.98 − 2.50i)2-s + (1.25 + 7.09i)4-s + (−2.58 + 14.6i)5-s + (5.35 + 9.26i)7-s + (−1.55 + 2.68i)8-s + (44.5 − 37.3i)10-s + (14.6 − 25.3i)11-s + (−76.4 − 27.8i)13-s + (7.24 − 41.0i)14-s + (65.5 − 23.8i)16-s + (−61.3 − 51.4i)17-s + (81.6 − 13.9i)19-s − 107.·20-s + (−107. + 39.0i)22-s + (−11.6 − 66.1i)23-s + ⋯ |
| L(s) = 1 | + (−1.05 − 0.886i)2-s + (0.156 + 0.886i)4-s + (−0.231 + 1.31i)5-s + (0.288 + 0.500i)7-s + (−0.0686 + 0.118i)8-s + (1.40 − 1.18i)10-s + (0.400 − 0.694i)11-s + (−1.63 − 0.593i)13-s + (0.138 − 0.784i)14-s + (1.02 − 0.372i)16-s + (−0.875 − 0.734i)17-s + (0.985 − 0.168i)19-s − 1.20·20-s + (−1.03 + 0.378i)22-s + (−0.105 − 0.599i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0242067 - 0.262322i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0242067 - 0.262322i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (-81.6 + 13.9i)T \) |
| good | 2 | \( 1 + (2.98 + 2.50i)T + (1.38 + 7.87i)T^{2} \) |
| 5 | \( 1 + (2.58 - 14.6i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (-5.35 - 9.26i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-14.6 + 25.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (76.4 + 27.8i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (61.3 + 51.4i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (11.6 + 66.1i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (1.78 - 1.49i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (68.9 + 119. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (213. - 77.8i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (29.9 - 169. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-324. + 271. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-28.7 - 163. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (355. + 298. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (106. + 603. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-166. + 139. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-9.51 + 53.9i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-130. + 47.5i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (436. - 158. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (43.2 + 74.9i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-443. - 161. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (326. + 274. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−11.55143229912141939742334126544, −10.81019990656158479946939171238, −9.931310278823146526149407594816, −9.030766289099546450896592285213, −7.85636629918059825495244952724, −6.85757275368275536274567263327, −5.27341941043724647113445423026, −3.16764217978753006928337663824, −2.29788865808723081430804279723, −0.17446798679723189038731488602,
1.45313197922347604859195985110, 4.18530951175117717460965412734, 5.29070440801668565623999398115, 6.95676159450839950539259386005, 7.59271072554618649658936233092, 8.742915639739477764200297889781, 9.362686441576193396085323560511, 10.32989719141397159530515933793, 11.98142463538909951163095430238, 12.53413702084497697789738148643
