| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 11-s − 13-s + 4·14-s + 16-s + 7·17-s − 3·19-s + 22-s + 26-s − 4·28-s + 4·29-s + 6·31-s − 32-s − 7·34-s − 8·37-s + 3·38-s + 5·41-s − 4·43-s − 44-s − 12·47-s + 9·49-s − 52-s + 10·53-s + 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 0.301·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.69·17-s − 0.688·19-s + 0.213·22-s + 0.196·26-s − 0.755·28-s + 0.742·29-s + 1.07·31-s − 0.176·32-s − 1.20·34-s − 1.31·37-s + 0.486·38-s + 0.780·41-s − 0.609·43-s − 0.150·44-s − 1.75·47-s + 9/7·49-s − 0.138·52-s + 1.37·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.935373015568568332918151417767, −6.91743915415021953631832658110, −6.56945747007592015604095334978, −5.76618054915883954286048327451, −4.99948608538217339043824418243, −3.77538938623471425734013588668, −3.13775484254017044407602024609, −2.39516618729307829982025671007, −1.09311391695805572816962115321, 0,
1.09311391695805572816962115321, 2.39516618729307829982025671007, 3.13775484254017044407602024609, 3.77538938623471425734013588668, 4.99948608538217339043824418243, 5.76618054915883954286048327451, 6.56945747007592015604095334978, 6.91743915415021953631832658110, 7.935373015568568332918151417767
