| L(s) = 1 | + (−0.451 + 1.34i)2-s + (−0.780 + 2.89i)3-s + (−1.59 − 1.21i)4-s + (2.36 + 0.941i)5-s + (−3.52 − 2.35i)6-s + (10.5 − 4.88i)7-s + (2.34 − 1.58i)8-s + (−7.78 − 4.51i)9-s + (−2.32 + 2.74i)10-s + (11.5 + 3.19i)11-s + (4.74 − 3.66i)12-s + (3.56 − 6.73i)13-s + (1.77 + 16.3i)14-s + (−4.56 + 6.10i)15-s + (1.07 + 3.85i)16-s + (12.8 − 27.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 + 0.670i)2-s + (−0.260 + 0.965i)3-s + (−0.398 − 0.302i)4-s + (0.472 + 0.188i)5-s + (−0.588 − 0.392i)6-s + (1.50 − 0.697i)7-s + (0.292 − 0.198i)8-s + (−0.864 − 0.502i)9-s + (−0.232 + 0.274i)10-s + (1.04 + 0.290i)11-s + (0.395 − 0.305i)12-s + (0.274 − 0.517i)13-s + (0.126 + 1.16i)14-s + (−0.304 + 0.407i)15-s + (0.0668 + 0.240i)16-s + (0.757 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.48350 + 0.955628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48350 + 0.955628i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.451 - 1.34i)T \) |
| 3 | \( 1 + (0.780 - 2.89i)T \) |
| 59 | \( 1 + (15.0 + 57.0i)T \) |
| good | 5 | \( 1 + (-2.36 - 0.941i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-10.5 + 4.88i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-11.5 - 3.19i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-3.56 + 6.73i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (-12.8 + 27.8i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-26.6 + 5.87i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (12.1 + 1.99i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (-14.4 - 42.9i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (15.4 + 3.40i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (33.9 - 50.0i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (37.6 - 6.16i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-6.49 - 23.3i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (58.7 - 23.4i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-46.8 + 39.7i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-0.376 - 0.126i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-34.6 - 51.1i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (50.1 - 19.9i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-10.2 + 1.11i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-71.5 + 43.0i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (75.7 - 4.10i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-4.10 - 12.1i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-108. - 11.8i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−11.38667800791183526490090342173, −10.24461907638217784360649465239, −9.685186626601340414224669414773, −8.656706444648403997529877304901, −7.65970914084133512443836129162, −6.64805276307548892664424956999, −5.27049946367128410435847321381, −4.82249186042128033425822126940, −3.43221383092800312712952978001, −1.17181827801793623994238650946,
1.40086649236111785815563232874, 1.92702945893164430273846732947, 3.78715282225070152339109371893, 5.35066746519190331617184846727, 6.04188344358684625185789165561, 7.55198567004126636132688128335, 8.351729645554260120766663639428, 9.075935303328782665534986454055, 10.32130064630864964177376554984, 11.50612600576227191105452484186
