| L(s) = 1 | − 4·2-s + 19.2·3-s + 16·4-s − 81.3·5-s − 76.8·6-s − 137.·7-s − 64·8-s + 126.·9-s + 325.·10-s − 458.·11-s + 307.·12-s − 1.04e3·13-s + 548.·14-s − 1.56e3·15-s + 256·16-s − 1.14e3·17-s − 505.·18-s + 1.26e3·19-s − 1.30e3·20-s − 2.63e3·21-s + 1.83e3·22-s + 2.58e3·23-s − 1.22e3·24-s + 3.49e3·25-s + 4.18e3·26-s − 2.24e3·27-s − 2.19e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.23·3-s + 0.5·4-s − 1.45·5-s − 0.871·6-s − 1.05·7-s − 0.353·8-s + 0.519·9-s + 1.02·10-s − 1.14·11-s + 0.616·12-s − 1.71·13-s + 0.747·14-s − 1.79·15-s + 0.250·16-s − 0.963·17-s − 0.367·18-s + 0.801·19-s − 0.727·20-s − 1.30·21-s + 0.807·22-s + 1.01·23-s − 0.435·24-s + 1.11·25-s + 1.21·26-s − 0.591·27-s − 0.528·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3284051114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3284051114\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 269 | \( 1 - 7.23e4T \) |
| good | 3 | \( 1 - 19.2T + 243T^{2} \) |
| 5 | \( 1 + 81.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 137.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 458.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.26e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.58e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.03e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.18e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.77e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.99e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.44e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.77e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.77e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.80e4T + 8.58e9T^{2} \) |
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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
−9.599191595366403425585757367439, −9.253342247224145437157107225909, −8.184289053608584052520068057962, −7.44077416596396879740088370490, −7.13355113049131521861183736048, −5.33636859563493964248740655653, −3.94967358938449034808501687080, −3.01585912728635301486886888670, −2.35549626940297673692731645523, −0.27214252504719193454276988243,
0.27214252504719193454276988243, 2.35549626940297673692731645523, 3.01585912728635301486886888670, 3.94967358938449034808501687080, 5.33636859563493964248740655653, 7.13355113049131521861183736048, 7.44077416596396879740088370490, 8.184289053608584052520068057962, 9.253342247224145437157107225909, 9.599191595366403425585757367439
