| L(s) = 1 | + 2.15·2-s + 2.43·3-s + 2.66·4-s − 5-s + 5.25·6-s + 7-s + 1.43·8-s + 2.93·9-s − 2.15·10-s + 0.840·11-s + 6.48·12-s − 4.09·13-s + 2.15·14-s − 2.43·15-s − 2.22·16-s + 5.15·17-s + 6.32·18-s + 4.38·19-s − 2.66·20-s + 2.43·21-s + 1.81·22-s − 23-s + 3.49·24-s + 25-s − 8.85·26-s − 0.169·27-s + 2.66·28-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.40·3-s + 1.33·4-s − 0.447·5-s + 2.14·6-s + 0.377·7-s + 0.507·8-s + 0.976·9-s − 0.682·10-s + 0.253·11-s + 1.87·12-s − 1.13·13-s + 0.577·14-s − 0.628·15-s − 0.557·16-s + 1.25·17-s + 1.49·18-s + 1.00·19-s − 0.595·20-s + 0.531·21-s + 0.386·22-s − 0.208·23-s + 0.713·24-s + 0.200·25-s − 1.73·26-s − 0.0325·27-s + 0.503·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.950955155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.950955155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 - 2.43T + 3T^{2} \) |
| 11 | \( 1 - 0.840T + 11T^{2} \) |
| 13 | \( 1 + 4.09T + 13T^{2} \) |
| 17 | \( 1 - 5.15T + 17T^{2} \) |
| 19 | \( 1 - 4.38T + 19T^{2} \) |
| 29 | \( 1 - 0.458T + 29T^{2} \) |
| 31 | \( 1 + 0.389T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 5.24T + 41T^{2} \) |
| 43 | \( 1 + 1.90T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 + 3.11T + 53T^{2} \) |
| 59 | \( 1 - 4.21T + 59T^{2} \) |
| 61 | \( 1 - 6.00T + 61T^{2} \) |
| 67 | \( 1 - 2.09T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 4.95T + 83T^{2} \) |
| 89 | \( 1 - 6.84T + 89T^{2} \) |
| 97 | \( 1 + 8.02T + 97T^{2} \) |
| 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
−10.18953770101807387117887029613, −9.386604780806954630346050435449, −8.384739735806814099808356560657, −7.59825643977048854499292611485, −6.86963738927671192581859535038, −5.45856466313702712422964586402, −4.75921271000124824013731056019, −3.58304833956471437153273121064, −3.16145045073289534222230379482, −1.97028288200708079191778777702,
1.97028288200708079191778777702, 3.16145045073289534222230379482, 3.58304833956471437153273121064, 4.75921271000124824013731056019, 5.45856466313702712422964586402, 6.86963738927671192581859535038, 7.59825643977048854499292611485, 8.384739735806814099808356560657, 9.386604780806954630346050435449, 10.18953770101807387117887029613
