| L(s) = 1 | − 2-s + 2.19·3-s + 4-s − 2.35·5-s − 2.19·6-s − 0.565·7-s − 8-s + 1.79·9-s + 2.35·10-s + 11-s + 2.19·12-s − 4.37·13-s + 0.565·14-s − 5.16·15-s + 16-s + 5.62·17-s − 1.79·18-s − 1.21·19-s − 2.35·20-s − 1.23·21-s − 22-s − 3.60·23-s − 2.19·24-s + 0.554·25-s + 4.37·26-s − 2.63·27-s − 0.565·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.26·3-s + 0.5·4-s − 1.05·5-s − 0.894·6-s − 0.213·7-s − 0.353·8-s + 0.599·9-s + 0.745·10-s + 0.301·11-s + 0.632·12-s − 1.21·13-s + 0.151·14-s − 1.33·15-s + 0.250·16-s + 1.36·17-s − 0.423·18-s − 0.279·19-s − 0.526·20-s − 0.270·21-s − 0.213·22-s − 0.751·23-s − 0.447·24-s + 0.110·25-s + 0.858·26-s − 0.506·27-s − 0.106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 7 | \( 1 + 0.565T + 7T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 5.62T + 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 + 3.60T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 + 5.16T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 3.77T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 9.29T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 7.18T + 97T^{2} \) |
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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
−8.143493651404025720872464538303, −7.62849974177563481714351138134, −6.90988700897842511253957576874, −6.03517531670198923791092420298, −4.80239766892848774578039224572, −4.00723651834487675977631567847, −3.09441268006644563195360493793, −2.65542704816865940859131242919, −1.40113026715751502222279800756, 0,
1.40113026715751502222279800756, 2.65542704816865940859131242919, 3.09441268006644563195360493793, 4.00723651834487675977631567847, 4.80239766892848774578039224572, 6.03517531670198923791092420298, 6.90988700897842511253957576874, 7.62849974177563481714351138134, 8.143493651404025720872464538303
