| L(s) = 1 | − 3·3-s + 10.5·5-s − 7·7-s + 9·9-s − 34.7·11-s − 37.2·13-s − 31.6·15-s − 10.5·17-s + 58.5·19-s + 21·21-s + 125.·23-s − 13.7·25-s − 27·27-s − 35.4·29-s − 291.·31-s + 104.·33-s − 73.8·35-s − 259.·37-s + 111.·39-s − 338.·41-s − 6.80·43-s + 94.9·45-s − 250.·47-s + 49·49-s + 31.6·51-s − 536.·53-s − 366.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.943·5-s − 0.377·7-s + 0.333·9-s − 0.952·11-s − 0.795·13-s − 0.544·15-s − 0.150·17-s + 0.707·19-s + 0.218·21-s + 1.13·23-s − 0.109·25-s − 0.192·27-s − 0.226·29-s − 1.69·31-s + 0.549·33-s − 0.356·35-s − 1.15·37-s + 0.459·39-s − 1.28·41-s − 0.0241·43-s + 0.314·45-s − 0.778·47-s + 0.142·49-s + 0.0868·51-s − 1.39·53-s − 0.898·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 10.5T + 125T^{2} \) |
| 11 | \( 1 + 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 35.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 6.80T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 35.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 57.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−10.55314252260507057026967221669, −9.851108601031503344667657035441, −9.035669858805245242311123142654, −7.59989857386556468797913429421, −6.73022719050793100184404417295, −5.56055115271339643577045628393, −4.98150113830503365689327347887, −3.20033352580597576395212986654, −1.82745178827283682517970098341, 0,
1.82745178827283682517970098341, 3.20033352580597576395212986654, 4.98150113830503365689327347887, 5.56055115271339643577045628393, 6.73022719050793100184404417295, 7.59989857386556468797913429421, 9.035669858805245242311123142654, 9.851108601031503344667657035441, 10.55314252260507057026967221669
