L(s) = 1 | + (−1.26 + 0.626i)2-s + (−1.49 − 2.59i)3-s + (1.21 − 1.58i)4-s + (−0.866 − 0.5i)5-s + (3.52 + 2.35i)6-s + (−2.06 + 1.65i)7-s + (−0.546 + 2.77i)8-s + (−2.99 + 5.18i)9-s + (1.41 + 0.0915i)10-s + (−1.93 + 1.11i)11-s + (−5.94 − 0.774i)12-s − 3.17i·13-s + (1.57 − 3.39i)14-s + 2.99i·15-s + (−1.04 − 3.86i)16-s + (−2.98 + 1.72i)17-s + ⋯ |
L(s) = 1 | + (−0.896 + 0.442i)2-s + (−0.865 − 1.49i)3-s + (0.607 − 0.794i)4-s + (−0.387 − 0.223i)5-s + (1.43 + 0.960i)6-s + (−0.778 + 0.627i)7-s + (−0.193 + 0.981i)8-s + (−0.998 + 1.72i)9-s + (0.446 + 0.0289i)10-s + (−0.584 + 0.337i)11-s + (−1.71 − 0.223i)12-s − 0.879i·13-s + (0.420 − 0.907i)14-s + 0.774i·15-s + (−0.261 − 0.965i)16-s + (−0.723 + 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00765284 + 0.149634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00765284 + 0.149634i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.26 - 0.626i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (2.06 - 1.65i)T \) |
good | 3 | \( 1 + (1.49 + 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (1.93 - 1.11i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.17iT - 13T^{2} \) |
| 17 | \( 1 + (2.98 - 1.72i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.02 + 1.77i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.30 + 1.33i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.38T + 29T^{2} \) |
| 31 | \( 1 + (2.44 + 4.23i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.59 + 9.69i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.46iT - 41T^{2} \) |
| 43 | \( 1 - 9.95iT - 43T^{2} \) |
| 47 | \( 1 + (-3.06 + 5.30i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.32 + 4.03i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.55 + 6.16i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.19 + 1.26i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0456 - 0.0263i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.212iT - 71T^{2} \) |
| 73 | \( 1 + (12.8 - 7.43i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.399 - 0.230i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + (-6.07 - 3.51i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.185iT - 97T^{2} \) |
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
−12.69057905032482277095778375899, −11.59163451437118349736578329423, −10.75050869188552129772753874656, −9.345971072780355651805879247689, −8.062425389919903135704703635283, −7.33739754806736776567927203235, −6.23425923114206488290601678133, −5.45412696717451772903600246428, −2.25683339619312794905118083572, −0.20797836147086459048649156517,
3.32215774306429371542578163452, 4.36362226369031652863186534553, 6.11218501147341045956209664328, 7.35640123044690191893539662311, 8.966388499891966365900703980012, 9.772049475057443294761962280998, 10.59576503400516880962270104005, 11.25489617566622851237682194195, 12.12969571674329907310539153184, 13.54462187034848327959602095121