| L(s) = 1 | + 0.759·2-s − 1.42·4-s − 4.29·7-s − 2.60·8-s + 1.35·11-s − 0.931·13-s − 3.26·14-s + 0.869·16-s + 3.18·17-s − 1.13·19-s + 1.02·22-s + 4.97·23-s − 0.708·26-s + 6.10·28-s + 4.66·29-s + 31-s + 5.86·32-s + 2.41·34-s − 10.9·37-s − 0.859·38-s + 11.4·41-s + 5.71·43-s − 1.92·44-s + 3.77·46-s + 0.503·47-s + 11.4·49-s + 1.32·52-s + ⋯ |
| L(s) = 1 | + 0.537·2-s − 0.711·4-s − 1.62·7-s − 0.919·8-s + 0.408·11-s − 0.258·13-s − 0.871·14-s + 0.217·16-s + 0.771·17-s − 0.259·19-s + 0.219·22-s + 1.03·23-s − 0.138·26-s + 1.15·28-s + 0.865·29-s + 0.179·31-s + 1.03·32-s + 0.414·34-s − 1.79·37-s − 0.139·38-s + 1.78·41-s + 0.871·43-s − 0.290·44-s + 0.556·46-s + 0.0734·47-s + 1.63·49-s + 0.183·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.759T + 2T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 + 0.931T + 13T^{2} \) |
| 17 | \( 1 - 3.18T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 - 4.66T + 29T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 5.71T + 43T^{2} \) |
| 47 | \( 1 - 0.503T + 47T^{2} \) |
| 53 | \( 1 - 2.52T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 4.27T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 2.28T + 79T^{2} \) |
| 83 | \( 1 + 8.83T + 83T^{2} \) |
| 89 | \( 1 + 1.21T + 89T^{2} \) |
| 97 | \( 1 + 9.32T + 97T^{2} \) |
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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
−7.42643260383062858902393411451, −6.79939967979078697969951041483, −6.05003544397528562731075968834, −5.57893883701497859592937216090, −4.64108003982407326475553078315, −3.97444019264737039518783982497, −3.18880556360606545139694703924, −2.74281628310662562436674913304, −1.09104300035880406158108776218, 0,
1.09104300035880406158108776218, 2.74281628310662562436674913304, 3.18880556360606545139694703924, 3.97444019264737039518783982497, 4.64108003982407326475553078315, 5.57893883701497859592937216090, 6.05003544397528562731075968834, 6.79939967979078697969951041483, 7.42643260383062858902393411451
