| L(s) = 1 | + 3-s − 2.37·5-s − 2.37·7-s + 9-s + 11-s − 13-s − 2.37·15-s − 4·17-s − 4.74·19-s − 2.37·21-s − 2.37·23-s + 0.627·25-s + 27-s + 2.37·29-s + 2.74·31-s + 33-s + 5.62·35-s − 2·37-s − 39-s − 0.372·41-s + 8.37·43-s − 2.37·45-s + 4·47-s − 1.37·49-s − 4·51-s − 10·53-s − 2.37·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.06·5-s − 0.896·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s − 0.612·15-s − 0.970·17-s − 1.08·19-s − 0.517·21-s − 0.494·23-s + 0.125·25-s + 0.192·27-s + 0.440·29-s + 0.492·31-s + 0.174·33-s + 0.951·35-s − 0.328·37-s − 0.160·39-s − 0.0581·41-s + 1.27·43-s − 0.353·45-s + 0.583·47-s − 0.196·49-s − 0.560·51-s − 1.37·53-s − 0.319·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.113359672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113359672\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 - 2.37T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 0.372T + 41T^{2} \) |
| 43 | \( 1 - 8.37T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 5.62T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 - 5.11T + 67T^{2} \) |
| 71 | \( 1 - 3.25T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 8.74T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.058508586655017691453983713457, −7.30973427211383251116458737653, −6.60289333364216568033246011246, −6.13836175738851683204064811013, −4.87070638132792079920474002832, −4.17025552249705982111939195818, −3.67569748612377254821245771520, −2.80438771709272235143884966835, −1.99979098434646542758184272738, −0.49852570388615041742254469647,
0.49852570388615041742254469647, 1.99979098434646542758184272738, 2.80438771709272235143884966835, 3.67569748612377254821245771520, 4.17025552249705982111939195818, 4.87070638132792079920474002832, 6.13836175738851683204064811013, 6.60289333364216568033246011246, 7.30973427211383251116458737653, 8.058508586655017691453983713457
