| L(s) = 1 | − 4.18·2-s − 14.4·4-s − 25·5-s + 86.0·7-s + 194.·8-s + 104.·10-s + 121·11-s + 47.8·13-s − 360.·14-s − 350.·16-s − 813.·17-s + 1.55e3·19-s + 362.·20-s − 506.·22-s + 4.11e3·23-s + 625·25-s − 200.·26-s − 1.24e3·28-s − 87.5·29-s − 6.32e3·31-s − 4.75e3·32-s + 3.40e3·34-s − 2.15e3·35-s + 1.56e4·37-s − 6.49e3·38-s − 4.86e3·40-s − 1.39e4·41-s + ⋯ |
| L(s) = 1 | − 0.739·2-s − 0.452·4-s − 0.447·5-s + 0.663·7-s + 1.07·8-s + 0.330·10-s + 0.301·11-s + 0.0784·13-s − 0.491·14-s − 0.342·16-s − 0.682·17-s + 0.986·19-s + 0.202·20-s − 0.223·22-s + 1.62·23-s + 0.200·25-s − 0.0580·26-s − 0.300·28-s − 0.0193·29-s − 1.18·31-s − 0.821·32-s + 0.505·34-s − 0.296·35-s + 1.87·37-s − 0.729·38-s − 0.480·40-s − 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.147462950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147462950\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 4.18T + 32T^{2} \) |
| 7 | \( 1 - 86.0T + 1.68e4T^{2} \) |
| 13 | \( 1 - 47.8T + 3.71e5T^{2} \) |
| 17 | \( 1 + 813.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 87.5T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.56e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.68e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.83e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.06e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.52e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 245.T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.99e4T + 8.58e9T^{2} \) |
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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
−9.964039506615505933703747282945, −9.180270210304059170490358844340, −8.439939569891704907132987777796, −7.65145435880967817571891509456, −6.79759871436496162450120309629, −5.23991385481874529364998340833, −4.52161365152369467012265299938, −3.32995173431265466852250245236, −1.66601303131317360112605435607, −0.64402586308749199583438925857,
0.64402586308749199583438925857, 1.66601303131317360112605435607, 3.32995173431265466852250245236, 4.52161365152369467012265299938, 5.23991385481874529364998340833, 6.79759871436496162450120309629, 7.65145435880967817571891509456, 8.439939569891704907132987777796, 9.180270210304059170490358844340, 9.964039506615505933703747282945
