| L(s) = 1 | − 1.73·2-s + 3-s + 0.999·4-s + 5-s − 1.73·6-s − 2.73·7-s + 1.73·8-s + 9-s − 1.73·10-s + 4.73·11-s + 0.999·12-s + 0.732·13-s + 4.73·14-s + 15-s − 5·16-s − 1.73·18-s + 19-s + 0.999·20-s − 2.73·21-s − 8.19·22-s + 3.46·23-s + 1.73·24-s + 25-s − 1.26·26-s + 27-s − 2.73·28-s + 8.19·29-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.577·3-s + 0.499·4-s + 0.447·5-s − 0.707·6-s − 1.03·7-s + 0.612·8-s + 0.333·9-s − 0.547·10-s + 1.42·11-s + 0.288·12-s + 0.203·13-s + 1.26·14-s + 0.258·15-s − 1.25·16-s − 0.408·18-s + 0.229·19-s + 0.223·20-s − 0.596·21-s − 1.74·22-s + 0.722·23-s + 0.353·24-s + 0.200·25-s − 0.248·26-s + 0.192·27-s − 0.516·28-s + 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8703069908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8703069908\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 - 0.732T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 8.19T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 6.53T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−11.66631361450594969104298348504, −10.40111776022432643504222121874, −9.726442139148524098460702342354, −9.064337119716004837733582986763, −8.357492743998930486544211870736, −7.03591498952022486703765867586, −6.33842567697968775115131274888, −4.46417869343420470917120240068, −3.01708959245831383050013519549, −1.27918120443323915706052052566,
1.27918120443323915706052052566, 3.01708959245831383050013519549, 4.46417869343420470917120240068, 6.33842567697968775115131274888, 7.03591498952022486703765867586, 8.357492743998930486544211870736, 9.064337119716004837733582986763, 9.726442139148524098460702342354, 10.40111776022432643504222121874, 11.66631361450594969104298348504
