| L(s) = 1 | + (−0.535 − 1.30i)2-s + (−1.45 + 0.946i)3-s + (−1.42 + 1.40i)4-s + (−2.23 − 0.138i)5-s + (2.01 + 1.39i)6-s + 1.71·7-s + (2.59 + 1.11i)8-s + (1.20 − 2.74i)9-s + (1.01 + 2.99i)10-s + (3.14 + 2.28i)11-s + (0.739 − 3.38i)12-s + (−2.67 − 3.68i)13-s + (−0.918 − 2.24i)14-s + (3.36 − 1.91i)15-s + (0.0644 − 3.99i)16-s + (1.44 − 4.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.378 − 0.925i)2-s + (−0.837 + 0.546i)3-s + (−0.712 + 0.701i)4-s + (−0.998 − 0.0617i)5-s + (0.823 + 0.567i)6-s + 0.648·7-s + (0.919 + 0.393i)8-s + (0.402 − 0.915i)9-s + (0.321 + 0.947i)10-s + (0.949 + 0.689i)11-s + (0.213 − 0.976i)12-s + (−0.741 − 1.02i)13-s + (−0.245 − 0.599i)14-s + (0.869 − 0.493i)15-s + (0.0161 − 0.999i)16-s + (0.350 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.495789 - 0.420496i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.495789 - 0.420496i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.535 + 1.30i)T \) |
| 3 | \( 1 + (1.45 - 0.946i)T \) |
| 5 | \( 1 + (2.23 + 0.138i)T \) |
| good | 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 + (-3.14 - 2.28i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.67 + 3.68i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 4.44i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.75 + 0.895i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.94 + 5.43i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-7.46 + 2.42i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.34 - 2.06i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.44 + 4.74i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.75 - 5.16i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.64T + 43T^{2} \) |
| 47 | \( 1 + (5.75 - 1.86i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.134 + 0.415i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.61 - 1.90i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.20 - 3.78i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.75 + 5.41i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (3.57 + 10.9i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.32 + 5.95i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (3.64 - 1.18i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.21 + 1.36i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (2.46 - 3.38i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-8.74 + 2.84i)T + (78.4 - 57.0i)T^{2} \) |
| 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
−11.53241545256305770431601678821, −10.71491982654817462891126846223, −9.920645055419840197143469805598, −8.893545454285680048689479655572, −7.86081736892235983099889357038, −6.79517787325536878294301692672, −4.82041241683233685843839015884, −4.50029505049749354047859140282, −2.99957464294226257994671276624, −0.74658816694846650432463831199,
1.28214560580435427203212720889, 4.11037459740421075325235856134, 5.04439945157926752407526998791, 6.34203982106728596033530360830, 7.01067498672140821126726801667, 8.025447273196086961758448474441, 8.710310841200772519894266737903, 10.11554537672026382310786507020, 11.17471241677466099988571280948, 11.78572725388596198016980721331
