| L(s) = 1 | + (2.72 + 3.41i)2-s + (−2.84 + 3.56i)3-s + (−2.46 + 10.7i)4-s + (8.30 − 3.99i)5-s − 19.9·6-s − 12.2·7-s + (−12.0 + 5.79i)8-s + (1.38 + 6.06i)9-s + (36.2 + 17.4i)10-s + (−3.74 − 16.3i)11-s + (−31.4 − 39.4i)12-s + (66.8 − 32.1i)13-s + (−33.3 − 41.8i)14-s + (−9.34 + 40.9i)15-s + (27.1 + 13.0i)16-s + (−0.640 − 0.308i)17-s + ⋯ |
| L(s) = 1 | + (0.962 + 1.20i)2-s + (−0.546 + 0.685i)3-s + (−0.307 + 1.34i)4-s + (0.742 − 0.357i)5-s − 1.35·6-s − 0.662·7-s + (−0.532 + 0.256i)8-s + (0.0512 + 0.224i)9-s + (1.14 + 0.551i)10-s + (−0.102 − 0.449i)11-s + (−0.756 − 0.948i)12-s + (1.42 − 0.686i)13-s + (−0.637 − 0.799i)14-s + (−0.160 + 0.704i)15-s + (0.424 + 0.204i)16-s + (−0.00914 − 0.00440i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.986525 + 1.60510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.986525 + 1.60510i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (163. + 229. i)T \) |
| good | 2 | \( 1 + (-2.72 - 3.41i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (2.84 - 3.56i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-8.30 + 3.99i)T + (77.9 - 97.7i)T^{2} \) |
| 7 | \( 1 + 12.2T + 343T^{2} \) |
| 11 | \( 1 + (3.74 + 16.3i)T + (-1.19e3 + 577. i)T^{2} \) |
| 13 | \( 1 + (-66.8 + 32.1i)T + (1.36e3 - 1.71e3i)T^{2} \) |
| 17 | \( 1 + (0.640 + 0.308i)T + (3.06e3 + 3.84e3i)T^{2} \) |
| 19 | \( 1 + (6.96 - 30.5i)T + (-6.17e3 - 2.97e3i)T^{2} \) |
| 23 | \( 1 + (18.6 + 81.7i)T + (-1.09e4 + 5.27e3i)T^{2} \) |
| 29 | \( 1 + (142. + 178. i)T + (-5.42e3 + 2.37e4i)T^{2} \) |
| 31 | \( 1 + (151. + 190. i)T + (-6.62e3 + 2.90e4i)T^{2} \) |
| 37 | \( 1 - 82.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-252. - 316. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 47 | \( 1 + (78.0 - 342. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (95.6 + 46.0i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (-616. - 296. i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-315. + 395. i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (63.7 - 279. i)T + (-2.70e5 - 1.30e5i)T^{2} \) |
| 71 | \( 1 + (-29.2 + 128. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (656. - 316. i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 - 818.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (540. - 677. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-545. + 684. i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 + (-207. - 908. i)T + (-8.22e5 + 3.95e5i)T^{2} \) |
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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
−16.02782697151167151763846939372, −14.87433423927075433336062743723, −13.42199807376489934479610208075, −13.05637461661929986939215786565, −11.05357290869439934029171984086, −9.747104830763258514669232680240, −8.036904355942193469112773623749, −6.14108082460607370624123776064, −5.55546952987834472459109408192, −3.96864602426520895272931004312,
1.69395511554357990536712200830, 3.61544787152034387085157469120, 5.65759924689113853986194375565, 6.82535222441984408684246470607, 9.392180624609604443132061309100, 10.70060288386595390824459573167, 11.69570766324802043825813057020, 12.82621577776243667587215205046, 13.41264471328674074070697833175, 14.54599200171755934514307302749
