| L(s) = 1 | + (−0.367 + 0.266i)2-s + (2.05 + 6.32i)3-s + (−2.40 + 7.41i)4-s + (4.33 + 3.14i)5-s + (−2.44 − 1.77i)6-s + (−8.57 + 26.4i)7-s + (−2.21 − 6.81i)8-s + (−13.9 + 10.1i)9-s − 2.42·10-s + (18.7 − 31.2i)11-s − 51.8·12-s + (10.5 − 7.64i)13-s + (−3.89 − 11.9i)14-s + (−11.0 + 33.8i)15-s + (−47.8 − 34.7i)16-s + (63.5 + 46.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.129 + 0.0942i)2-s + (0.395 + 1.21i)3-s + (−0.301 + 0.926i)4-s + (0.387 + 0.281i)5-s + (−0.166 − 0.120i)6-s + (−0.463 + 1.42i)7-s + (−0.0978 − 0.301i)8-s + (−0.517 + 0.376i)9-s − 0.0768·10-s + (0.514 − 0.857i)11-s − 1.24·12-s + (0.224 − 0.163i)13-s + (−0.0743 − 0.228i)14-s + (−0.189 + 0.583i)15-s + (−0.747 − 0.542i)16-s + (0.906 + 0.658i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.195332 + 1.55773i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.195332 + 1.55773i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-18.7 + 31.2i)T \) |
| 13 | \( 1 + (-10.5 + 7.64i)T \) |
| good | 2 | \( 1 + (0.367 - 0.266i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (-2.05 - 6.32i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-4.33 - 3.14i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (8.57 - 26.4i)T + (-277. - 201. i)T^{2} \) |
| 17 | \( 1 + (-63.5 - 46.1i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (22.6 + 69.7i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 47.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + (21.5 - 66.1i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (221. - 160. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (72.5 - 223. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-95.6 - 294. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 375.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (155. + 478. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (316. - 229. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-228. + 702. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-735. - 534. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 235.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (272. + 198. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-145. + 449. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (117. - 85.4i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-962. - 699. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 354.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (570. - 414. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.03337571210961817677254361994, −12.15923437879847034509973411384, −10.96943150196922072435398628624, −9.724432956517888542468615665325, −8.974330113951209955677790505001, −8.359546796884771683055191756337, −6.54536235568831517461102526498, −5.25369889677155299759857894948, −3.70031911426273061271693510903, −2.86579237231793622032363562189,
0.821163291465854030792421698164, 1.85227094849078268242630312919, 4.05828737547275132592681254316, 5.70683322814162574543523866555, 6.94055250472262096750220007272, 7.65312981449847398031275943987, 9.247125224239891467144518320267, 9.946943521152131798751547876695, 11.06739887417543920430157419522, 12.54342390437103214342873018167
