| L(s) = 1 | + 0.561·2-s + 3·3-s − 7.68·4-s + 18.6·5-s + 1.68·6-s + 7·7-s − 8.80·8-s + 9·9-s + 10.4·10-s − 11·11-s − 23.0·12-s + 36.4·13-s + 3.93·14-s + 56.0·15-s + 56.5·16-s + 41.1·17-s + 5.05·18-s − 23.6·19-s − 143.·20-s + 21·21-s − 6.17·22-s − 140.·23-s − 26.4·24-s + 224.·25-s + 20.4·26-s + 27·27-s − 53.7·28-s + ⋯ |
| L(s) = 1 | + 0.198·2-s + 0.577·3-s − 0.960·4-s + 1.67·5-s + 0.114·6-s + 0.377·7-s − 0.389·8-s + 0.333·9-s + 0.331·10-s − 0.301·11-s − 0.554·12-s + 0.777·13-s + 0.0750·14-s + 0.964·15-s + 0.883·16-s + 0.586·17-s + 0.0661·18-s − 0.286·19-s − 1.60·20-s + 0.218·21-s − 0.0598·22-s − 1.26·23-s − 0.224·24-s + 1.79·25-s + 0.154·26-s + 0.192·27-s − 0.363·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.708015857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.708015857\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 0.561T + 8T^{2} \) |
| 5 | \( 1 - 18.6T + 125T^{2} \) |
| 13 | \( 1 - 36.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 3.67T + 7.95e4T^{2} \) |
| 47 | \( 1 + 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 597.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 668.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 667.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 31.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 434.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 782.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 426.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 899.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 942.T + 9.12e5T^{2} \) |
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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
−11.98919978424583024658441025743, −10.22086137157344132895095267758, −9.975513788543823733855051199593, −8.778654827260416413432319458108, −8.161433505919597216027701969169, −6.39369662888452053872925597664, −5.50172407959130252243984210451, −4.37035250923888642440931641446, −2.82463910437626107578622233504, −1.35609486101962426245410304782,
1.35609486101962426245410304782, 2.82463910437626107578622233504, 4.37035250923888642440931641446, 5.50172407959130252243984210451, 6.39369662888452053872925597664, 8.161433505919597216027701969169, 8.778654827260416413432319458108, 9.975513788543823733855051199593, 10.22086137157344132895095267758, 11.98919978424583024658441025743
