| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 61.8·5-s − 36·6-s + 49·7-s − 64·8-s + 81·9-s + 247.·10-s + 121·11-s + 144·12-s − 36.0·13-s − 196·14-s − 556.·15-s + 256·16-s − 76.6·17-s − 324·18-s + 2.78e3·19-s − 989.·20-s + 441·21-s − 484·22-s − 4.57e3·23-s − 576·24-s + 701.·25-s + 144.·26-s + 729·27-s + 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.10·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.782·10-s + 0.301·11-s + 0.288·12-s − 0.0591·13-s − 0.267·14-s − 0.638·15-s + 0.250·16-s − 0.0643·17-s − 0.235·18-s + 1.76·19-s − 0.553·20-s + 0.218·21-s − 0.213·22-s − 1.80·23-s − 0.204·24-s + 0.224·25-s + 0.0417·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.452558029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.452558029\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 - 49T \) |
| 11 | \( 1 - 121T \) |
| good | 5 | \( 1 + 61.8T + 3.12e3T^{2} \) |
| 13 | \( 1 + 36.0T + 3.71e5T^{2} \) |
| 17 | \( 1 + 76.6T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.78e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.62e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.85e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.62e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.61e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.12e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.59e4T + 8.58e9T^{2} \) |
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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
−10.05969236855182593611643137161, −9.305148615296999931286116060418, −8.282183204455919535732010967581, −7.74543509974456797351202558463, −7.00286940019357639944771677670, −5.58975766755886693506405401874, −4.17986115396981440930176490878, −3.32172409263404671984003832120, −1.95726275474324577766465642938, −0.66959743088157491939817927842,
0.66959743088157491939817927842, 1.95726275474324577766465642938, 3.32172409263404671984003832120, 4.17986115396981440930176490878, 5.58975766755886693506405401874, 7.00286940019357639944771677670, 7.74543509974456797351202558463, 8.282183204455919535732010967581, 9.305148615296999931286116060418, 10.05969236855182593611643137161
