| L(s) = 1 | − 2·5-s + 5.12·11-s + 4.56·13-s + 3.12·17-s − 5.12·19-s − 23-s − 25-s + 0.561·29-s + 6.56·31-s − 8.24·37-s − 10.8·41-s + 8·43-s + 11.6·47-s − 7·49-s − 2·53-s − 10.2·55-s − 6.24·59-s + 12.2·61-s − 9.12·65-s + 5.12·67-s + 9.43·71-s − 2.31·73-s + 5.12·79-s − 2.24·83-s − 6.24·85-s + 13.3·89-s + 10.2·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.54·11-s + 1.26·13-s + 0.757·17-s − 1.17·19-s − 0.208·23-s − 0.200·25-s + 0.104·29-s + 1.17·31-s − 1.35·37-s − 1.68·41-s + 1.21·43-s + 1.70·47-s − 49-s − 0.274·53-s − 1.38·55-s − 0.813·59-s + 1.56·61-s − 1.13·65-s + 0.625·67-s + 1.12·71-s − 0.270·73-s + 0.576·79-s − 0.246·83-s − 0.677·85-s + 1.41·89-s + 1.05·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.763241098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.763241098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 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
−8.484652865828331613722270960161, −8.116282381550409061528746723874, −7.03716308732209282300110852364, −6.46915736737043775846941667305, −5.75026148095619163971865868535, −4.56005859389813946471320411952, −3.85227798450954084882438326413, −3.38799005375643845830239346399, −1.89323272175238904203744384988, −0.826726240466976182437078614861,
0.826726240466976182437078614861, 1.89323272175238904203744384988, 3.38799005375643845830239346399, 3.85227798450954084882438326413, 4.56005859389813946471320411952, 5.75026148095619163971865868535, 6.46915736737043775846941667305, 7.03716308732209282300110852364, 8.116282381550409061528746723874, 8.484652865828331613722270960161
