| L(s) = 1 | − 2.56·3-s + 2·5-s + 3.56·9-s + 5.12·11-s + 4.56·13-s − 5.12·15-s − 3.12·17-s + 5.12·19-s − 23-s − 25-s − 1.43·27-s − 0.561·29-s − 6.56·31-s − 13.1·33-s − 8.24·37-s − 11.6·39-s + 10.8·41-s − 8·43-s + 7.12·45-s + 11.6·47-s − 7·49-s + 8·51-s + 2·53-s + 10.2·55-s − 13.1·57-s − 6.24·59-s + 12.2·61-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 0.894·5-s + 1.18·9-s + 1.54·11-s + 1.26·13-s − 1.32·15-s − 0.757·17-s + 1.17·19-s − 0.208·23-s − 0.200·25-s − 0.276·27-s − 0.104·29-s − 1.17·31-s − 2.28·33-s − 1.35·37-s − 1.87·39-s + 1.68·41-s − 1.21·43-s + 1.06·45-s + 1.70·47-s − 49-s + 1.12·51-s + 0.274·53-s + 1.38·55-s − 1.73·57-s − 0.813·59-s + 1.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9308842755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9308842755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 29 | \( 1 + 0.561T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 - 9.43T + 71T^{2} \) |
| 73 | \( 1 + 2.31T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−12.42166299949270058109746642113, −11.50360440708631699291947275935, −10.91275619474228902607669351037, −9.743313128799501782362351196014, −8.847535098094654825696978388046, −6.98921704429773607482381407808, −6.15229576525720049532558019035, −5.43391800843212204758872795626, −3.93675101810801057119345153741, −1.41817274069713329556021365021,
1.41817274069713329556021365021, 3.93675101810801057119345153741, 5.43391800843212204758872795626, 6.15229576525720049532558019035, 6.98921704429773607482381407808, 8.847535098094654825696978388046, 9.743313128799501782362351196014, 10.91275619474228902607669351037, 11.50360440708631699291947275935, 12.42166299949270058109746642113
