| L(s) = 1 | − 4·2-s − 11.0·3-s + 16·4-s − 52.2·5-s + 44.1·6-s + 215.·7-s − 64·8-s − 120.·9-s + 208.·10-s + 546.·11-s − 176.·12-s − 257.·13-s − 862.·14-s + 576.·15-s + 256·16-s + 1.89e3·17-s + 483.·18-s − 835.·20-s − 2.38e3·21-s − 2.18e3·22-s + 2.23e3·23-s + 706.·24-s − 398.·25-s + 1.03e3·26-s + 4.02e3·27-s + 3.45e3·28-s + 3.58e3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.708·3-s + 0.5·4-s − 0.933·5-s + 0.501·6-s + 1.66·7-s − 0.353·8-s − 0.497·9-s + 0.660·10-s + 1.36·11-s − 0.354·12-s − 0.423·13-s − 1.17·14-s + 0.661·15-s + 0.250·16-s + 1.59·17-s + 0.352·18-s − 0.466·20-s − 1.17·21-s − 0.963·22-s + 0.880·23-s + 0.250·24-s − 0.127·25-s + 0.299·26-s + 1.06·27-s + 0.831·28-s + 0.791·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.419555512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419555512\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 11.0T + 243T^{2} \) |
| 5 | \( 1 + 52.2T + 3.12e3T^{2} \) |
| 7 | \( 1 - 215.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 546.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 257.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.89e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.25e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.02e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.59e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.06e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.77e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.43e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.57e5T + 8.58e9T^{2} \) |
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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
−9.650503640352761850886485723682, −8.538057744485376250936344211861, −8.050262277978061444347176173793, −7.23612076844614931932261913302, −6.19208454439431391820920641475, −5.14044652191775092491073049724, −4.30655826378156756109755332163, −2.99339275930673444391986860086, −1.40601273948562599713562337768, −0.73227629584693673288206640923,
0.73227629584693673288206640923, 1.40601273948562599713562337768, 2.99339275930673444391986860086, 4.30655826378156756109755332163, 5.14044652191775092491073049724, 6.19208454439431391820920641475, 7.23612076844614931932261913302, 8.050262277978061444347176173793, 8.538057744485376250936344211861, 9.650503640352761850886485723682
