| L(s) = 1 | − 2.67·3-s + 1.73·5-s + 1.26·7-s + 4.14·9-s + 11-s + 0.398·13-s − 4.64·15-s + 4.77·17-s − 3.52·19-s − 3.39·21-s − 6.70·23-s − 1.98·25-s − 3.06·27-s + 2.17·29-s − 2.14·31-s − 2.67·33-s + 2.20·35-s − 9.51·37-s − 1.06·39-s − 4.25·41-s + 3.23·43-s + 7.20·45-s − 7.48·47-s − 5.38·49-s − 12.7·51-s + 53-s + 1.73·55-s + ⋯ |
| L(s) = 1 | − 1.54·3-s + 0.776·5-s + 0.479·7-s + 1.38·9-s + 0.301·11-s + 0.110·13-s − 1.19·15-s + 1.15·17-s − 0.809·19-s − 0.740·21-s − 1.39·23-s − 0.396·25-s − 0.589·27-s + 0.403·29-s − 0.384·31-s − 0.465·33-s + 0.372·35-s − 1.56·37-s − 0.170·39-s − 0.664·41-s + 0.493·43-s + 1.07·45-s − 1.09·47-s − 0.769·49-s − 1.78·51-s + 0.137·53-s + 0.234·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 2.67T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 13 | \( 1 - 0.398T + 13T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 9.51T + 37T^{2} \) |
| 41 | \( 1 + 4.25T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 59 | \( 1 + 5.82T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 6.18T + 67T^{2} \) |
| 71 | \( 1 - 3.77T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 5.77T + 79T^{2} \) |
| 83 | \( 1 + 7.77T + 83T^{2} \) |
| 89 | \( 1 + 8.96T + 89T^{2} \) |
| 97 | \( 1 - 2.58T + 97T^{2} \) |
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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
−7.931280763015582346931550522042, −6.92348531076278807654948329581, −6.35513229972647147815903710595, −5.69028146461684806870186611654, −5.25726948873677870919807696865, −4.41974326651912719994785484564, −3.51792503376517893215404657209, −2.05449762586867988418354985150, −1.32747074739325918742946017171, 0,
1.32747074739325918742946017171, 2.05449762586867988418354985150, 3.51792503376517893215404657209, 4.41974326651912719994785484564, 5.25726948873677870919807696865, 5.69028146461684806870186611654, 6.35513229972647147815903710595, 6.92348531076278807654948329581, 7.931280763015582346931550522042
