| L(s) = 1 | + 1.92·3-s + 0.603·5-s − 4.00·7-s + 0.702·9-s + 2.54·11-s + 0.384·13-s + 1.16·15-s − 4.88·17-s − 0.138·19-s − 7.70·21-s + 4.63·23-s − 4.63·25-s − 4.42·27-s + 0.641·29-s − 0.324·31-s + 4.88·33-s − 2.41·35-s + 2.71·37-s + 0.739·39-s − 11.1·41-s − 5.21·43-s + 0.423·45-s − 2.12·47-s + 9.01·49-s − 9.39·51-s − 3.74·53-s + 1.53·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 0.269·5-s − 1.51·7-s + 0.234·9-s + 0.765·11-s + 0.106·13-s + 0.299·15-s − 1.18·17-s − 0.0318·19-s − 1.68·21-s + 0.966·23-s − 0.927·25-s − 0.850·27-s + 0.119·29-s − 0.0582·31-s + 0.851·33-s − 0.407·35-s + 0.445·37-s + 0.118·39-s − 1.73·41-s − 0.795·43-s + 0.0631·45-s − 0.309·47-s + 1.28·49-s − 1.31·51-s − 0.514·53-s + 0.206·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 1.92T + 3T^{2} \) |
| 5 | \( 1 - 0.603T + 5T^{2} \) |
| 7 | \( 1 + 4.00T + 7T^{2} \) |
| 11 | \( 1 - 2.54T + 11T^{2} \) |
| 13 | \( 1 - 0.384T + 13T^{2} \) |
| 17 | \( 1 + 4.88T + 17T^{2} \) |
| 19 | \( 1 + 0.138T + 19T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 - 0.641T + 29T^{2} \) |
| 31 | \( 1 + 0.324T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 + 2.12T + 47T^{2} \) |
| 53 | \( 1 + 3.74T + 53T^{2} \) |
| 59 | \( 1 + 6.32T + 59T^{2} \) |
| 61 | \( 1 + 7.81T + 61T^{2} \) |
| 67 | \( 1 + 3.82T + 67T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 - 1.50T + 73T^{2} \) |
| 79 | \( 1 - 2.50T + 79T^{2} \) |
| 83 | \( 1 - 8.03T + 83T^{2} \) |
| 89 | \( 1 - 0.594T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
| 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
−8.275765839852498099903993498234, −7.32199201697768692716840458983, −6.54758287467043139085741342432, −6.18208251572257567405208070021, −5.00764008559497305606723417030, −3.93420469353615448225434597744, −3.33675961602317576771429768214, −2.64555782655818501480037179199, −1.67250431315552910192768142669, 0,
1.67250431315552910192768142669, 2.64555782655818501480037179199, 3.33675961602317576771429768214, 3.93420469353615448225434597744, 5.00764008559497305606723417030, 6.18208251572257567405208070021, 6.54758287467043139085741342432, 7.32199201697768692716840458983, 8.275765839852498099903993498234
