| L(s) = 1 | + (−1.41 + 0.0249i)2-s + (−1.51 − 0.301i)3-s + (1.99 − 0.0705i)4-s + (1.90 − 1.27i)5-s + (2.14 + 0.387i)6-s + (2.53 − 3.79i)7-s + (−2.82 + 0.149i)8-s + (−0.571 − 0.236i)9-s + (−2.66 + 1.85i)10-s + (0.737 + 3.70i)11-s + (−3.04 − 0.494i)12-s + (1.06 + 1.06i)13-s + (−3.48 + 5.42i)14-s + (−3.27 + 1.35i)15-s + (3.99 − 0.282i)16-s + (−3.21 − 2.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0176i)2-s + (−0.873 − 0.173i)3-s + (0.999 − 0.0352i)4-s + (0.853 − 0.570i)5-s + (0.876 + 0.158i)6-s + (0.957 − 1.43i)7-s + (−0.998 + 0.0529i)8-s + (−0.190 − 0.0789i)9-s + (−0.843 + 0.585i)10-s + (0.222 + 1.11i)11-s + (−0.879 − 0.142i)12-s + (0.296 + 0.296i)13-s + (−0.932 + 1.44i)14-s + (−0.845 + 0.350i)15-s + (0.997 − 0.0705i)16-s + (−0.779 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.520859 - 0.214305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.520859 - 0.214305i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0249i)T \) |
| 17 | \( 1 + (3.21 + 2.58i)T \) |
| good | 3 | \( 1 + (1.51 + 0.301i)T + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-1.90 + 1.27i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-2.53 + 3.79i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.737 - 3.70i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 1.06i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.308 - 0.745i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.55 - 0.310i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.05 - 1.58i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (0.416 - 2.09i)T + (-28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (1.44 - 7.27i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-6.06 - 4.05i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.146 + 0.352i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (0.465 - 0.465i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.45 + 1.01i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (5.44 + 2.25i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (3.92 - 5.87i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + 16.2T + 67T^{2} \) |
| 71 | \( 1 + (-7.36 - 1.46i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-5.84 + 3.90i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (2.33 + 11.7i)T + (-72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (-6.25 + 2.58i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-5.52 + 5.52i)T - 89iT^{2} \) |
| 97 | \( 1 + (-3.39 - 5.07i)T + (-37.1 + 89.6i)T^{2} \) |
| 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
−14.71420106783982098646816801307, −13.53489064429796617671822462338, −12.08615365355207320183099039624, −11.12953210913370861943188600429, −10.18829701484952609777604880344, −9.016834867824573991682563177154, −7.51385808623818504879125737049, −6.41856843589276254702038110342, −4.83303884933705100038503831165, −1.44375205495162125440148858811,
2.36832519406400931961134308438, 5.66651539471928831147833361887, 6.19748952378691146245603493026, 8.226861910876183682532151352848, 9.133117362743938011073849057965, 10.69782353451495956231459469501, 11.20085187346036909477326592863, 12.22268087736603725050397734256, 14.05415381749113296976681783476, 15.18535837902726527106287051095
