L(s) = 1 | + (1.26 + 2.18i)3-s + (−0.5 − 0.866i)5-s + 2.72·7-s + (−1.67 + 2.90i)9-s + 3.31·11-s + (−1.62 + 2.81i)13-s + (1.26 − 2.18i)15-s + (1.17 + 2.03i)17-s + (−3.11 − 3.04i)19-s + (3.43 + 5.95i)21-s + (−1.07 + 1.86i)23-s + (−0.499 + 0.866i)25-s − 0.893·27-s + (1.96 − 3.40i)29-s − 10.1·31-s + ⋯ |
L(s) = 1 | + (0.727 + 1.26i)3-s + (−0.223 − 0.387i)5-s + 1.03·7-s + (−0.559 + 0.968i)9-s + 0.999·11-s + (−0.450 + 0.780i)13-s + (0.325 − 0.563i)15-s + (0.285 + 0.494i)17-s + (−0.714 − 0.699i)19-s + (0.750 + 1.29i)21-s + (−0.223 + 0.387i)23-s + (−0.0999 + 0.173i)25-s − 0.171·27-s + (0.365 − 0.632i)29-s − 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.54095 + 0.952012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54095 + 0.952012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (3.11 + 3.04i)T \) |
good | 3 | \( 1 + (-1.26 - 2.18i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 13 | \( 1 + (1.62 - 2.81i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.17 - 2.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (1.07 - 1.86i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.96 + 3.40i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 + (-0.363 - 0.629i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.18 - 2.05i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.51 + 9.54i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.49 + 7.77i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.48 - 9.50i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.22 + 7.32i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.87 + 8.44i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.45 + 5.99i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.24 - 2.14i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.99 + 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 + (4.27 - 7.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.61 + 6.26i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.40363297963930455595563687520, −10.54438836845327964768214834849, −9.489747012537142359488760140733, −8.895520703592685193183815762988, −8.144675283000637328687940240142, −6.90400435701199192527706506818, −5.32233138334260588054563674682, −4.36485583685468051003612318099, −3.73248915548215278393516785151, −1.96908588031302520275192266506,
1.39795602157506891481188536356, 2.58104200211940855445101081780, 3.96582561127663072115051724034, 5.48707127908677488243485917875, 6.77631177304371483819394490817, 7.49180250503373422809292603406, 8.227332066461852114191053030253, 9.034699477021568230936051014680, 10.39106828550345252261150883214, 11.34243626168811471846588523610