| L(s) = 1 | − 0.561·2-s − 3·3-s − 7.68·4-s + 1.68·6-s − 7·7-s + 8.80·8-s + 9·9-s − 25.8·11-s + 23.0·12-s − 15.5·13-s + 3.93·14-s + 56.5·16-s − 95.1·17-s − 5.05·18-s − 143.·19-s + 21·21-s + 14.4·22-s + 77.5·23-s − 26.4·24-s + 8.73·26-s − 27·27-s + 53.7·28-s − 204.·29-s − 40.6·31-s − 102.·32-s + 77.4·33-s + 53.4·34-s + ⋯ |
| L(s) = 1 | − 0.198·2-s − 0.577·3-s − 0.960·4-s + 0.114·6-s − 0.377·7-s + 0.389·8-s + 0.333·9-s − 0.707·11-s + 0.554·12-s − 0.331·13-s + 0.0750·14-s + 0.883·16-s − 1.35·17-s − 0.0661·18-s − 1.73·19-s + 0.218·21-s + 0.140·22-s + 0.702·23-s − 0.224·24-s + 0.0658·26-s − 0.192·27-s + 0.363·28-s − 1.30·29-s − 0.235·31-s − 0.564·32-s + 0.408·33-s + 0.269·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4868373993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4868373993\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 0.561T + 8T^{2} \) |
| 11 | \( 1 + 25.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 77.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 95.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 24.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 282.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 257.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 257.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 651.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 451.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 832.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 174.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 47.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.49e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.36e3T + 9.12e5T^{2} \) |
| 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
−10.59333444576037829179880456064, −9.468960479219499458394271749213, −8.849056421123131864142375321339, −7.81572400732461855890984495501, −6.77388928272800678244653833131, −5.72902192319760416974702302077, −4.75123181023720967039259971988, −3.93060693162802408963263995972, −2.25968941286165932325369206806, −0.43840591368660462215669846199,
0.43840591368660462215669846199, 2.25968941286165932325369206806, 3.93060693162802408963263995972, 4.75123181023720967039259971988, 5.72902192319760416974702302077, 6.77388928272800678244653833131, 7.81572400732461855890984495501, 8.849056421123131864142375321339, 9.468960479219499458394271749213, 10.59333444576037829179880456064
