| L(s) = 1 | − 6.40·2-s + 9.01·4-s − 25·5-s − 122.·7-s + 147.·8-s + 160.·10-s − 121·11-s − 1.04e3·13-s + 783.·14-s − 1.23e3·16-s − 400.·17-s + 581.·19-s − 225.·20-s + 774.·22-s − 66.9·23-s + 625·25-s + 6.67e3·26-s − 1.10e3·28-s − 6.78e3·29-s − 3.86e3·31-s + 3.17e3·32-s + 2.56e3·34-s + 3.05e3·35-s − 1.45e4·37-s − 3.72e3·38-s − 3.67e3·40-s + 5.66e3·41-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.281·4-s − 0.447·5-s − 0.944·7-s + 0.813·8-s + 0.506·10-s − 0.301·11-s − 1.71·13-s + 1.06·14-s − 1.20·16-s − 0.336·17-s + 0.369·19-s − 0.126·20-s + 0.341·22-s − 0.0263·23-s + 0.200·25-s + 1.93·26-s − 0.266·28-s − 1.49·29-s − 0.721·31-s + 0.548·32-s + 0.380·34-s + 0.422·35-s − 1.74·37-s − 0.418·38-s − 0.363·40-s + 0.526·41-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
| \(L(3)\) |
\(\approx\) |
\(0.04934005976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04934005976\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 6.40T + 32T^{2} \) |
| 7 | \( 1 + 122.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 400.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 581.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 66.9T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.45e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.85e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.73e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.46e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.99e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−9.892755468272771542295454049454, −9.392478835986795011980187540430, −8.482494813909725353361446695559, −7.39670208394240131590225291879, −7.04466568621221623682653665355, −5.48530437706610915984378841669, −4.40172319885019867758844243939, −3.11045479584836308618155666269, −1.81152927641800510336424798837, −0.12832639702705327969625943948,
0.12832639702705327969625943948, 1.81152927641800510336424798837, 3.11045479584836308618155666269, 4.40172319885019867758844243939, 5.48530437706610915984378841669, 7.04466568621221623682653665355, 7.39670208394240131590225291879, 8.482494813909725353361446695559, 9.392478835986795011980187540430, 9.892755468272771542295454049454
