| L(s) = 1 | + (−1.18 + 0.774i)2-s + (2.55 − 1.28i)3-s + (0.800 − 1.83i)4-s + (−0.374 − 0.845i)5-s + (−2.02 + 3.50i)6-s + (2.13 + 1.27i)7-s + (0.472 + 2.78i)8-s + (3.09 − 4.17i)9-s + (1.09 + 0.710i)10-s + (0.772 + 0.602i)11-s + (−0.313 − 5.71i)12-s + (0.634 − 0.604i)13-s + (−3.51 + 0.139i)14-s + (−2.04 − 1.68i)15-s + (−2.71 − 2.93i)16-s + (3.03 + 3.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.836 + 0.547i)2-s + (1.47 − 0.743i)3-s + (0.400 − 0.916i)4-s + (−0.167 − 0.378i)5-s + (−0.828 + 1.43i)6-s + (0.805 + 0.483i)7-s + (0.167 + 0.985i)8-s + (1.03 − 1.39i)9-s + (0.347 + 0.224i)10-s + (0.232 + 0.181i)11-s + (−0.0904 − 1.65i)12-s + (0.175 − 0.167i)13-s + (−0.938 + 0.0372i)14-s + (−0.528 − 0.433i)15-s + (−0.679 − 0.733i)16-s + (0.735 + 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67958 - 0.248519i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67958 - 0.248519i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.18 - 0.774i)T \) |
| good | 3 | \( 1 + (-2.55 + 1.28i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (0.374 + 0.845i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-2.13 - 1.27i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-0.772 - 0.602i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.634 + 0.604i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-3.03 - 3.69i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-0.501 - 2.88i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (5.48 + 2.59i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (5.68 + 3.21i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-2.47 + 0.492i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (2.86 + 1.81i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-2.03 - 2.24i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (3.13 + 9.50i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (3.06 + 1.63i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (6.60 - 3.73i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (5.20 - 5.47i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.715 - 9.69i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-9.38 - 8.09i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-7.65 - 1.13i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-6.53 + 3.91i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (-2.30 - 0.699i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (1.91 - 1.21i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-1.09 + 0.516i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (16.1 + 10.7i)T + (37.1 + 89.6i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.52855605298083579117199462345, −9.624205992571048012744009888455, −8.698051837868623344422520457150, −8.161540056002026990395335324720, −7.74341058072407310771151971115, −6.57482637615782026754324572386, −5.49108707188057807137228099826, −3.94919427519225338976196887784, −2.32735851867664591298017371864, −1.43054756643175523385160908388,
1.67474265786674275163186023777, 3.01672707361029539344904414449, 3.69574993870704905957202509341, 4.83474240276115126194416272639, 6.88741937164557028411530101092, 7.84260942435054104883211050081, 8.244724049599109296359821141609, 9.483252024902288986430238083166, 9.629433373246763241274180521510, 10.92684655520062201230430802093
