| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 9.05·5-s − 6·6-s − 16.7·7-s + 8·8-s + 9·9-s + 18.1·10-s + 16.8·11-s − 12·12-s − 29.6·13-s − 33.4·14-s − 27.1·15-s + 16·16-s + 68.2·17-s + 18·18-s + 36.2·20-s + 50.1·21-s + 33.7·22-s − 193.·23-s − 24·24-s − 43.0·25-s − 59.3·26-s − 27·27-s − 66.8·28-s + 69.5·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.809·5-s − 0.408·6-s − 0.902·7-s + 0.353·8-s + 0.333·9-s + 0.572·10-s + 0.463·11-s − 0.288·12-s − 0.633·13-s − 0.638·14-s − 0.467·15-s + 0.250·16-s + 0.973·17-s + 0.235·18-s + 0.404·20-s + 0.521·21-s + 0.327·22-s − 1.75·23-s − 0.204·24-s − 0.344·25-s − 0.447·26-s − 0.192·27-s − 0.451·28-s + 0.445·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 9.05T + 125T^{2} \) |
| 7 | \( 1 + 16.7T + 343T^{2} \) |
| 11 | \( 1 - 16.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 193.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 77.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 193.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 423.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 114.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 618.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 217.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 342.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 539.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 557.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 590.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 221.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 215.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 592.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 101.T + 9.12e5T^{2} \) |
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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
−8.199040792802191064946714301559, −7.29985626201921623271681221055, −6.41183274363266999441730145399, −6.01345455671121279365462259830, −5.28187079903579109963736958503, −4.31339781880772009450912937205, −3.43034590084142440511131217115, −2.41829351834550457881546097532, −1.39373221407634778492872813124, 0,
1.39373221407634778492872813124, 2.41829351834550457881546097532, 3.43034590084142440511131217115, 4.31339781880772009450912937205, 5.28187079903579109963736958503, 6.01345455671121279365462259830, 6.41183274363266999441730145399, 7.29985626201921623271681221055, 8.199040792802191064946714301559
