| L(s) = 1 | + 1.98·2-s + 1.31·3-s + 1.93·4-s + 5-s + 2.60·6-s − 2.40·7-s − 0.129·8-s − 1.27·9-s + 1.98·10-s − 11-s + 2.54·12-s − 4.30·13-s − 4.77·14-s + 1.31·15-s − 4.12·16-s − 6.12·17-s − 2.52·18-s + 3.06·19-s + 1.93·20-s − 3.16·21-s − 1.98·22-s + 1.49·23-s − 0.170·24-s + 25-s − 8.54·26-s − 5.61·27-s − 4.66·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.759·3-s + 0.967·4-s + 0.447·5-s + 1.06·6-s − 0.910·7-s − 0.0458·8-s − 0.423·9-s + 0.627·10-s − 0.301·11-s + 0.734·12-s − 1.19·13-s − 1.27·14-s + 0.339·15-s − 1.03·16-s − 1.48·17-s − 0.593·18-s + 0.702·19-s + 0.432·20-s − 0.691·21-s − 0.422·22-s + 0.311·23-s − 0.0347·24-s + 0.200·25-s − 1.67·26-s − 1.08·27-s − 0.880·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 3 | \( 1 - 1.31T + 3T^{2} \) |
| 7 | \( 1 + 2.40T + 7T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 + 6.12T + 17T^{2} \) |
| 19 | \( 1 - 3.06T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 + 0.652T + 31T^{2} \) |
| 37 | \( 1 - 4.83T + 37T^{2} \) |
| 41 | \( 1 + 6.80T + 41T^{2} \) |
| 43 | \( 1 - 3.66T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 6.21T + 53T^{2} \) |
| 59 | \( 1 + 6.95T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 79 | \( 1 + 1.91T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 6.44T + 89T^{2} \) |
| 97 | \( 1 + 5.45T + 97T^{2} \) |
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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
−8.095206478407696066752930045499, −6.97128229287759914680509949356, −6.56472197276877213481385419756, −5.71394101508541309875579892385, −4.97040654267955787377498160598, −4.33321731243760579327479199200, −3.19556837847955863988548709417, −2.86058747842388871719610512578, −2.10523428535301125231070022715, 0,
2.10523428535301125231070022715, 2.86058747842388871719610512578, 3.19556837847955863988548709417, 4.33321731243760579327479199200, 4.97040654267955787377498160598, 5.71394101508541309875579892385, 6.56472197276877213481385419756, 6.97128229287759914680509949356, 8.095206478407696066752930045499
