| L(s) = 1 | + 1.73·2-s + 0.999·4-s − 0.732·7-s − 1.73·8-s − 1.26·11-s + 2.73·13-s − 1.26·14-s − 5·16-s + 19-s − 2.19·22-s − 3.46·23-s + 4.73·26-s − 0.732·28-s + 2.19·29-s − 4.92·31-s − 5.19·32-s − 4.19·37-s + 1.73·38-s − 4.73·41-s + 6.19·43-s − 1.26·44-s − 5.99·46-s + 3.46·47-s − 6.46·49-s + 2.73·52-s − 2.53·53-s + 1.26·56-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s − 0.276·7-s − 0.612·8-s − 0.382·11-s + 0.757·13-s − 0.338·14-s − 1.25·16-s + 0.229·19-s − 0.468·22-s − 0.722·23-s + 0.928·26-s − 0.138·28-s + 0.407·29-s − 0.885·31-s − 0.918·32-s − 0.689·37-s + 0.280·38-s − 0.739·41-s + 0.944·43-s − 0.191·44-s − 0.884·46-s + 0.505·47-s − 0.923·49-s + 0.378·52-s − 0.348·53-s + 0.169·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 + 4.19T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 - 6.19T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 3.07T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 0.928T + 83T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 + 4.19T + 97T^{2} \) |
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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
−7.923128585609495901131717403728, −7.12118694406023396604155730121, −6.20161234007806772985896294760, −5.82854235818431139571929139487, −4.96023856434070909229650463352, −4.26491004000706411604043663440, −3.46100367701697235820858514210, −2.84936533175766301720700473225, −1.66002056075804824891594858009, 0,
1.66002056075804824891594858009, 2.84936533175766301720700473225, 3.46100367701697235820858514210, 4.26491004000706411604043663440, 4.96023856434070909229650463352, 5.82854235818431139571929139487, 6.20161234007806772985896294760, 7.12118694406023396604155730121, 7.923128585609495901131717403728
