| L(s) = 1 | + (−0.730 − 1.86i)2-s + (0.443 + 5.92i)3-s + (−2.93 + 2.72i)4-s + (3.03 − 2.06i)5-s + (10.7 − 5.15i)6-s + (−16.2 + 8.93i)7-s + (7.20 + 3.47i)8-s + (−8.16 + 1.23i)9-s + (−6.06 − 4.13i)10-s + (−24.7 − 3.72i)11-s + (−17.4 − 16.1i)12-s + (−50.9 + 63.9i)13-s + (28.4 + 23.6i)14-s + (13.5 + 17.0i)15-s + (1.19 − 15.9i)16-s + (−83.9 + 25.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.258 − 0.658i)2-s + (0.0854 + 1.13i)3-s + (−0.366 + 0.340i)4-s + (0.271 − 0.184i)5-s + (0.728 − 0.350i)6-s + (−0.875 + 0.482i)7-s + (0.318 + 0.153i)8-s + (−0.302 + 0.0455i)9-s + (−0.191 − 0.130i)10-s + (−0.677 − 0.102i)11-s + (−0.418 − 0.388i)12-s + (−1.08 + 1.36i)13-s + (0.543 + 0.451i)14-s + (0.233 + 0.293i)15-s + (0.0186 − 0.249i)16-s + (−1.19 + 0.369i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.404812 + 0.686035i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.404812 + 0.686035i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.730 + 1.86i)T \) |
| 7 | \( 1 + (16.2 - 8.93i)T \) |
| good | 3 | \( 1 + (-0.443 - 5.92i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (-3.03 + 2.06i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (24.7 + 3.72i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (50.9 - 63.9i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (83.9 - 25.8i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-3.16 + 5.48i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-190. - 58.6i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-38.1 + 167. i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (11.5 + 19.9i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (154. + 143. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (100. + 48.5i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-316. + 152. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-68.5 - 174. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (175. - 162. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-685. - 467. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-266. - 247. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-164. - 284. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-51.4 - 225. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-256. + 653. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-159. + 276. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-456. - 572. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (348. - 52.5i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + 1.37e3T + 9.12e5T^{2} \) |
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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
−13.56633637397475185370355050520, −12.69176282312444318437711675771, −11.45362869934825279752921673767, −10.41871497164194948264275114943, −9.375751968067314305634533210280, −9.040147244458862135792354427923, −7.04660926391900736045592808908, −5.23320596780373766932824724746, −4.02417926069888115927491345128, −2.47298681683031868396839815145,
0.47158799383148899012528465856, 2.68608310216615147950624957701, 5.06900094988513749277325248765, 6.63362852790266482605553113095, 7.18964954500938402605152456876, 8.325344234226314936785921847747, 9.776597330492967889096459958050, 10.70370979896745929892888470159, 12.61862604662697164354208085702, 13.00917549521167827549651002367
