| L(s) = 1 | − 2.16·2-s + 2.70·4-s + 5-s − 0.484·7-s − 1.53·8-s − 2.16·10-s − 2.30·11-s + 6.79·13-s + 1.05·14-s − 2.08·16-s − 4.79·17-s + 3.23·19-s + 2.70·20-s + 4.99·22-s + 3.55·23-s + 25-s − 14.7·26-s − 1.31·28-s − 29-s + 0.766·31-s + 7.59·32-s + 10.4·34-s − 0.484·35-s + 0.474·37-s − 7.01·38-s − 1.53·40-s + 2.45·41-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.35·4-s + 0.447·5-s − 0.183·7-s − 0.541·8-s − 0.686·10-s − 0.694·11-s + 1.88·13-s + 0.281·14-s − 0.522·16-s − 1.16·17-s + 0.741·19-s + 0.605·20-s + 1.06·22-s + 0.741·23-s + 0.200·25-s − 2.89·26-s − 0.248·28-s − 0.185·29-s + 0.137·31-s + 1.34·32-s + 1.78·34-s − 0.0819·35-s + 0.0780·37-s − 1.13·38-s − 0.242·40-s + 0.382·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8206558538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8206558538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 7 | \( 1 + 0.484T + 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 - 6.79T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 - 3.55T + 23T^{2} \) |
| 31 | \( 1 - 0.766T + 31T^{2} \) |
| 37 | \( 1 - 0.474T + 37T^{2} \) |
| 41 | \( 1 - 2.45T + 41T^{2} \) |
| 43 | \( 1 - 9.42T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 - 4.67T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 + 6.75T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 2.46T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 1.37T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 1.22T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−9.457472946476541778319002893185, −8.910056851265056053263962879695, −8.271355666069337877296558297651, −7.41304393312225337235619136808, −6.55704058694539461677275565022, −5.79779870103423653197644060499, −4.54302315878970876482830109445, −3.17695605084658500160973344360, −1.99076997122069926951900168881, −0.857855472269903162865508309743,
0.857855472269903162865508309743, 1.99076997122069926951900168881, 3.17695605084658500160973344360, 4.54302315878970876482830109445, 5.79779870103423653197644060499, 6.55704058694539461677275565022, 7.41304393312225337235619136808, 8.271355666069337877296558297651, 8.910056851265056053263962879695, 9.457472946476541778319002893185
