| L(s) = 1 | + 1.97·2-s + 1.51·3-s + 1.88·4-s + 5-s + 2.98·6-s − 7-s − 0.228·8-s − 0.703·9-s + 1.97·10-s − 3.02·11-s + 2.85·12-s + 3.27·13-s − 1.97·14-s + 1.51·15-s − 4.21·16-s − 4.51·17-s − 1.38·18-s − 0.240·19-s + 1.88·20-s − 1.51·21-s − 5.96·22-s − 0.872·23-s − 0.346·24-s + 25-s + 6.45·26-s − 5.61·27-s − 1.88·28-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 0.874·3-s + 0.942·4-s + 0.447·5-s + 1.21·6-s − 0.377·7-s − 0.0808·8-s − 0.234·9-s + 0.623·10-s − 0.913·11-s + 0.824·12-s + 0.908·13-s − 0.526·14-s + 0.391·15-s − 1.05·16-s − 1.09·17-s − 0.326·18-s − 0.0552·19-s + 0.421·20-s − 0.330·21-s − 1.27·22-s − 0.181·23-s − 0.0707·24-s + 0.200·25-s + 1.26·26-s − 1.08·27-s − 0.356·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.97T + 2T^{2} \) |
| 3 | \( 1 - 1.51T + 3T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 13 | \( 1 - 3.27T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 + 0.240T + 19T^{2} \) |
| 23 | \( 1 + 0.872T + 23T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 + 6.38T + 41T^{2} \) |
| 43 | \( 1 + 0.526T + 43T^{2} \) |
| 47 | \( 1 - 0.375T + 47T^{2} \) |
| 53 | \( 1 - 0.715T + 53T^{2} \) |
| 59 | \( 1 - 9.95T + 59T^{2} \) |
| 61 | \( 1 - 7.72T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 1.95T + 71T^{2} \) |
| 73 | \( 1 - 1.71T + 73T^{2} \) |
| 79 | \( 1 + 6.99T + 79T^{2} \) |
| 83 | \( 1 - 4.88T + 83T^{2} \) |
| 89 | \( 1 + 8.49T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
| 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
−7.22856990750603908130205745879, −6.67421789840684425312139126805, −5.87069151487712642518231171541, −5.40611151034612133068832561684, −4.63346763406175394109540880678, −3.73603048601782843087290074905, −3.26937637205977470726616996251, −2.51310445689158375836132989026, −1.87095114801973888734268423128, 0,
1.87095114801973888734268423128, 2.51310445689158375836132989026, 3.26937637205977470726616996251, 3.73603048601782843087290074905, 4.63346763406175394109540880678, 5.40611151034612133068832561684, 5.87069151487712642518231171541, 6.67421789840684425312139126805, 7.22856990750603908130205745879
