| L(s) = 1 | − 8.81·2-s + 28.7·3-s + 45.6·4-s − 253.·6-s − 98.9·7-s − 120.·8-s + 585.·9-s + 107.·11-s + 1.31e3·12-s − 345.·13-s + 872.·14-s − 398.·16-s − 1.40e3·17-s − 5.16e3·18-s − 138.·19-s − 2.84e3·21-s − 945.·22-s − 3.55e3·23-s − 3.47e3·24-s + 3.04e3·26-s + 9.86e3·27-s − 4.52e3·28-s − 1.51e3·29-s + 1.55e3·31-s + 7.37e3·32-s + 3.08e3·33-s + 1.23e4·34-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 1.84·3-s + 1.42·4-s − 2.87·6-s − 0.763·7-s − 0.666·8-s + 2.41·9-s + 0.267·11-s + 2.63·12-s − 0.566·13-s + 1.18·14-s − 0.388·16-s − 1.17·17-s − 3.75·18-s − 0.0878·19-s − 1.40·21-s − 0.416·22-s − 1.40·23-s − 1.23·24-s + 0.883·26-s + 2.60·27-s − 1.08·28-s − 0.334·29-s + 0.290·31-s + 1.27·32-s + 0.493·33-s + 1.83·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.626489121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.626489121\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 8.81T + 32T^{2} \) |
| 3 | \( 1 - 28.7T + 243T^{2} \) |
| 7 | \( 1 + 98.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 107.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 345.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.40e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 138.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 192.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 129.T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.38e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.05e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.15e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.47e4T + 8.58e9T^{2} \) |
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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
−9.021442713009379195364710853485, −8.578851339212609207105127006521, −7.77360853432398169592292716280, −7.14623234812426016282851649876, −6.35926191412758565658556529166, −4.45968951055146007359282322441, −3.55488752354349461542679121561, −2.39598138446202115569549422108, −1.96898645424664469056492584253, −0.61075150762260216997560426561,
0.61075150762260216997560426561, 1.96898645424664469056492584253, 2.39598138446202115569549422108, 3.55488752354349461542679121561, 4.45968951055146007359282322441, 6.35926191412758565658556529166, 7.14623234812426016282851649876, 7.77360853432398169592292716280, 8.578851339212609207105127006521, 9.021442713009379195364710853485
