| L(s) = 1 | − 5.54·2-s + 22.7·4-s − 5·5-s + 17.7·7-s − 81.4·8-s + 27.7·10-s − 11·11-s − 53.5·13-s − 98.2·14-s + 269.·16-s + 112.·17-s − 1.24·19-s − 113.·20-s + 60.9·22-s − 78.4·23-s + 25·25-s + 296.·26-s + 402.·28-s + 174.·29-s + 82.5·31-s − 842.·32-s − 622.·34-s − 88.6·35-s − 149.·37-s + 6.89·38-s + 407.·40-s + 414.·41-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 2.83·4-s − 0.447·5-s + 0.957·7-s − 3.60·8-s + 0.876·10-s − 0.301·11-s − 1.14·13-s − 1.87·14-s + 4.21·16-s + 1.60·17-s − 0.0150·19-s − 1.26·20-s + 0.590·22-s − 0.710·23-s + 0.200·25-s + 2.23·26-s + 2.71·28-s + 1.11·29-s + 0.478·31-s − 4.65·32-s − 3.14·34-s − 0.428·35-s − 0.662·37-s + 0.0294·38-s + 1.60·40-s + 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6988104468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6988104468\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 5.54T + 8T^{2} \) |
| 7 | \( 1 - 17.7T + 343T^{2} \) |
| 13 | \( 1 + 53.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.24T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 82.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 438.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 6.02T + 2.05e5T^{2} \) |
| 61 | \( 1 - 434.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 935.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 226.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 825.T + 9.12e5T^{2} \) |
| 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
−10.20672237730589418456899443548, −9.788862677689159843487528334864, −8.599860879493931278595111134271, −7.87145637950891353029742098856, −7.50324635499564904570436148445, −6.30928913985966804371390680267, −5.02283511744375275728796674552, −3.11693164324586899965869499479, −1.91498623203931473346507171641, −0.69486889885444194589707473589,
0.69486889885444194589707473589, 1.91498623203931473346507171641, 3.11693164324586899965869499479, 5.02283511744375275728796674552, 6.30928913985966804371390680267, 7.50324635499564904570436148445, 7.87145637950891353029742098856, 8.599860879493931278595111134271, 9.788862677689159843487528334864, 10.20672237730589418456899443548
