L(s) = 1 | + 5·5-s − 4·7-s + 72·11-s − 6·13-s − 38·17-s − 52·19-s + 152·23-s + 25·25-s + 78·29-s − 120·31-s − 20·35-s − 150·37-s − 362·41-s + 484·43-s + 280·47-s − 327·49-s + 670·53-s + 360·55-s + 696·59-s + 222·61-s − 30·65-s + 4·67-s + 96·71-s + 178·73-s − 288·77-s + 632·79-s − 612·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.215·7-s + 1.97·11-s − 0.128·13-s − 0.542·17-s − 0.627·19-s + 1.37·23-s + 1/5·25-s + 0.499·29-s − 0.695·31-s − 0.0965·35-s − 0.666·37-s − 1.37·41-s + 1.71·43-s + 0.868·47-s − 0.953·49-s + 1.73·53-s + 0.882·55-s + 1.53·59-s + 0.465·61-s − 0.0572·65-s + 0.00729·67-s + 0.160·71-s + 0.285·73-s − 0.426·77-s + 0.900·79-s − 0.809·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.424086190\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.424086190\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 72 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 + 120 T + p^{3} T^{2} \) |
| 37 | \( 1 + 150 T + p^{3} T^{2} \) |
| 41 | \( 1 + 362 T + p^{3} T^{2} \) |
| 43 | \( 1 - 484 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 - 670 T + p^{3} T^{2} \) |
| 59 | \( 1 - 696 T + p^{3} T^{2} \) |
| 61 | \( 1 - 222 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 178 T + p^{3} T^{2} \) |
| 79 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 + 994 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1634 T + p^{3} T^{2} \) |
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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
−9.927879345687842411253188442127, −9.040359145508973512920696820811, −8.655331802412648001115257485215, −7.06287814534809035049817351253, −6.64789042668547664881110348912, −5.61173594436924589927459403878, −4.44214354041109251000216738143, −3.51235449494208983025107828017, −2.13660290910793640930210806454, −0.933719639472869479412692177614,
0.933719639472869479412692177614, 2.13660290910793640930210806454, 3.51235449494208983025107828017, 4.44214354041109251000216738143, 5.61173594436924589927459403878, 6.64789042668547664881110348912, 7.06287814534809035049817351253, 8.655331802412648001115257485215, 9.040359145508973512920696820811, 9.927879345687842411253188442127