| L(s) = 1 | + (−1.17 + 1.61i)2-s + (−9.37 + 3.04i)3-s + (−1.23 − 3.80i)4-s + (9.19 − 6.35i)5-s + (6.09 − 18.7i)6-s − 2.82i·7-s + (7.60 + 2.47i)8-s + (56.8 − 41.2i)9-s + (−0.526 + 22.3i)10-s + (−39.2 − 28.5i)11-s + (23.1 + 31.9i)12-s + (−14.4 − 19.8i)13-s + (4.57 + 3.32i)14-s + (−66.8 + 87.6i)15-s + (−12.9 + 9.40i)16-s + (−33.4 − 10.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 + 0.572i)2-s + (−1.80 + 0.586i)3-s + (−0.154 − 0.475i)4-s + (0.822 − 0.568i)5-s + (0.414 − 1.27i)6-s − 0.152i·7-s + (0.336 + 0.109i)8-s + (2.10 − 1.52i)9-s + (−0.0166 + 0.706i)10-s + (−1.07 − 0.781i)11-s + (0.557 + 0.767i)12-s + (−0.307 − 0.423i)13-s + (0.0874 + 0.0635i)14-s + (−1.15 + 1.50i)15-s + (−0.202 + 0.146i)16-s + (−0.477 − 0.155i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.378160 - 0.235843i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378160 - 0.235843i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 - 1.61i)T \) |
| 5 | \( 1 + (-9.19 + 6.35i)T \) |
| good | 3 | \( 1 + (9.37 - 3.04i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 + 2.82iT - 343T^{2} \) |
| 11 | \( 1 + (39.2 + 28.5i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (14.4 + 19.8i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (33.4 + 10.8i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-35.2 + 108. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-4.87 + 6.70i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-37.5 - 115. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (17.2 - 53.0i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (256. + 352. i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (263. - 191. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 294. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-212. + 69.2i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (255. - 82.8i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (325. - 236. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (241. + 175. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-778. - 253. i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-36.0 - 111. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (168. - 231. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-3.64 - 11.2i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-67.0 - 21.7i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-835. - 606. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-665. + 216. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−15.52898101812727299676918819972, −13.61332997601741153990152750360, −12.47756007393206882777554024242, −10.97454639657720955568613335930, −10.27393923174747066962507739775, −9.012458172488970538248070812702, −6.96046273335489454439197890329, −5.61950020063810650794556494089, −4.96395714959949849094647307431, −0.47556807440481346837704908425,
1.88137054747663476351356454528, 5.03616651431093879398088303560, 6.34369628490518963143948741155, 7.56471863680924758511862345286, 9.933074475692165696604708091904, 10.55841634520602096718486176909, 11.73459672564247711475869126421, 12.61457156990853087719881199014, 13.64628694954377419400914114784, 15.58339334134334254748511025605
