| L(s) = 1 | − 0.270·2-s − 7.92·4-s − 33.4·7-s + 4.31·8-s − 11·11-s − 33.5·13-s + 9.04·14-s + 62.2·16-s + 71.1·17-s − 48.9·19-s + 2.97·22-s − 66.7·23-s + 9.07·26-s + 264.·28-s + 66.8·29-s + 145.·31-s − 51.3·32-s − 19.2·34-s − 37.8·37-s + 13.2·38-s + 344.·41-s + 34.4·43-s + 87.1·44-s + 18.0·46-s + 270.·47-s + 773.·49-s + 265.·52-s + ⋯ |
| L(s) = 1 | − 0.0957·2-s − 0.990·4-s − 1.80·7-s + 0.190·8-s − 0.301·11-s − 0.715·13-s + 0.172·14-s + 0.972·16-s + 1.01·17-s − 0.591·19-s + 0.0288·22-s − 0.605·23-s + 0.0684·26-s + 1.78·28-s + 0.428·29-s + 0.844·31-s − 0.283·32-s − 0.0972·34-s − 0.168·37-s + 0.0566·38-s + 1.31·41-s + 0.122·43-s + 0.298·44-s + 0.0579·46-s + 0.840·47-s + 2.25·49-s + 0.708·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 0.270T + 8T^{2} \) |
| 7 | \( 1 + 33.4T + 343T^{2} \) |
| 13 | \( 1 + 33.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 66.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 145.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 37.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 344.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 34.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 270.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 666.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 876.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 783.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 876.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 523.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 91.0T + 3.89e5T^{2} \) |
| 79 | \( 1 + 96.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 644.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.209739264685198307458831128850, −7.55787224371006000160003688583, −6.54681525046414131509362250683, −5.91442296481857221772142656375, −5.03665990600615723866433710799, −4.09422400284768962739412733198, −3.34296729543635956596358844498, −2.49374129924513318313799864129, −0.834678551434502142470249187270, 0,
0.834678551434502142470249187270, 2.49374129924513318313799864129, 3.34296729543635956596358844498, 4.09422400284768962739412733198, 5.03665990600615723866433710799, 5.91442296481857221772142656375, 6.54681525046414131509362250683, 7.55787224371006000160003688583, 8.209739264685198307458831128850
