| L(s) = 1 | − 1.50·2-s − 5.73·4-s + 0.155·5-s − 7·7-s + 20.6·8-s − 0.233·10-s + 11·11-s + 88.7·13-s + 10.5·14-s + 14.7·16-s − 102.·17-s − 26.8·19-s − 0.890·20-s − 16.5·22-s − 65.1·23-s − 124.·25-s − 133.·26-s + 40.1·28-s − 136.·29-s − 87.8·31-s − 187.·32-s + 153.·34-s − 1.08·35-s + 391.·37-s + 40.4·38-s + 3.20·40-s + 69.8·41-s + ⋯ |
| L(s) = 1 | − 0.532·2-s − 0.716·4-s + 0.0138·5-s − 0.377·7-s + 0.913·8-s − 0.00739·10-s + 0.301·11-s + 1.89·13-s + 0.201·14-s + 0.230·16-s − 1.45·17-s − 0.324·19-s − 0.00995·20-s − 0.160·22-s − 0.590·23-s − 0.999·25-s − 1.00·26-s + 0.270·28-s − 0.874·29-s − 0.508·31-s − 1.03·32-s + 0.775·34-s − 0.00524·35-s + 1.73·37-s + 0.172·38-s + 0.0126·40-s + 0.266·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.038997353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038997353\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 1.50T + 8T^{2} \) |
| 5 | \( 1 - 0.155T + 125T^{2} \) |
| 13 | \( 1 - 88.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 65.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 87.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 391.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 69.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 293.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 122.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 140.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 653.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 295.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 82.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 579.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 420.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 59.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 280.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 19.1T + 9.12e5T^{2} \) |
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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
−9.865219278374624669867951696012, −9.137723879258319793347386106311, −8.532911566815249268145027707022, −7.68655387436397038856427235205, −6.46996501872286748473572509047, −5.72917317210509237914354272474, −4.28388123289865148516018119376, −3.73443347680024143159591878249, −1.97199235079222243830832791133, −0.64850170131321244074605860951,
0.64850170131321244074605860951, 1.97199235079222243830832791133, 3.73443347680024143159591878249, 4.28388123289865148516018119376, 5.72917317210509237914354272474, 6.46996501872286748473572509047, 7.68655387436397038856427235205, 8.532911566815249268145027707022, 9.137723879258319793347386106311, 9.865219278374624669867951696012
