L(s) = 1 | + (0.866 + 0.5i)2-s + (−1.73 + i)3-s + (0.499 + 0.866i)4-s + (2.12 + 0.686i)5-s − 1.99·6-s + (2.00 − 1.15i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (1.5 + 1.65i)10-s + 4.31·11-s + (−1.73 − 0.999i)12-s + (−3.46 + 2i)13-s + 2.31·14-s + (−4.37 + 0.939i)15-s + (−0.5 + 0.866i)16-s + (−4.60 − 2.65i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.999 + 0.577i)3-s + (0.249 + 0.433i)4-s + (0.951 + 0.306i)5-s − 0.816·6-s + (0.758 − 0.437i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.474 + 0.524i)10-s + 1.30·11-s + (−0.499 − 0.288i)12-s + (−0.960 + 0.554i)13-s + 0.619·14-s + (−1.12 + 0.242i)15-s + (−0.125 + 0.216i)16-s + (−1.11 − 0.644i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0823 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0823 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24447 + 1.14582i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24447 + 1.14582i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-2.12 - 0.686i)T \) |
| 37 | \( 1 + (-2.59 + 5.5i)T \) |
good | 3 | \( 1 + (1.73 - i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-2.00 + 1.15i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 + (3.46 - 2i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (4.60 + 2.65i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.15 - 3.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 41 | \( 1 + (3.81 + 6.61i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 8.94iT - 43T^{2} \) |
| 47 | \( 1 - 4.31iT - 47T^{2} \) |
| 53 | \( 1 + (-7.47 - 4.31i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.31 + 10.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.97 + 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.45 + 0.841i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.84 - 6.65i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2.63iT - 73T^{2} \) |
| 79 | \( 1 + (5.47 + 9.48i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.74 - 3.31i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.81 + 10.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.94iT - 97T^{2} \) |
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
−11.52537850034187015408210688052, −10.95945443276275621818888162153, −9.864716421650195814941422565956, −9.088244821552297638866288962910, −7.47209426271322226984421287013, −6.64298679679868742657955775327, −5.64530715403442425148328128217, −4.89849127946528970334212524799, −3.90661864910432203478654490584, −1.97308957337219426172340879660,
1.22325277491720704754375292076, 2.49877187507108648673795629662, 4.51497283382260931273053777047, 5.29138227166719596708249517742, 6.27567235695224819592929744217, 6.85995393994479590304409287551, 8.514588824413218247022982482442, 9.475267746424306893921298760088, 10.54263182017105035420949528164, 11.53218156817976270789322870202