| L(s) = 1 | + 2.47·2-s + 3·3-s − 1.87·4-s − 7.81·5-s + 7.42·6-s + 8.05·7-s − 24.4·8-s + 9·9-s − 19.3·10-s + 40.4·11-s − 5.62·12-s − 41.5·13-s + 19.9·14-s − 23.4·15-s − 45.4·16-s + 63.0·17-s + 22.2·18-s + 40.0·19-s + 14.6·20-s + 24.1·21-s + 100.·22-s + 23·23-s − 73.3·24-s − 63.8·25-s − 102.·26-s + 27·27-s − 15.1·28-s + ⋯ |
| L(s) = 1 | + 0.874·2-s + 0.577·3-s − 0.234·4-s − 0.699·5-s + 0.505·6-s + 0.435·7-s − 1.08·8-s + 0.333·9-s − 0.611·10-s + 1.10·11-s − 0.135·12-s − 0.887·13-s + 0.380·14-s − 0.403·15-s − 0.710·16-s + 0.899·17-s + 0.291·18-s + 0.483·19-s + 0.163·20-s + 0.251·21-s + 0.969·22-s + 0.208·23-s − 0.623·24-s − 0.510·25-s − 0.776·26-s + 0.192·27-s − 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.285263334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.285263334\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 2.47T + 8T^{2} \) |
| 5 | \( 1 + 7.81T + 125T^{2} \) |
| 7 | \( 1 - 8.05T + 343T^{2} \) |
| 11 | \( 1 - 40.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.0T + 6.85e3T^{2} \) |
| 31 | \( 1 - 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 406.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 10.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 317.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 349.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 545.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 392.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 253.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 786.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 932.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 689.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 994.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 908.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−8.661371586338088846275680435045, −8.104780443721432020131103595616, −7.21684329307486914810516264642, −6.43150185503363606993929443672, −5.24857386571492835221935605341, −4.74038735050286606050755600267, −3.68608511704562642653988973519, −3.39161563708656299507045485107, −2.05581251163804225542527084217, −0.71461271862694620555693736235,
0.71461271862694620555693736235, 2.05581251163804225542527084217, 3.39161563708656299507045485107, 3.68608511704562642653988973519, 4.74038735050286606050755600267, 5.24857386571492835221935605341, 6.43150185503363606993929443672, 7.21684329307486914810516264642, 8.104780443721432020131103595616, 8.661371586338088846275680435045
