| L(s) = 1 | + 2·2-s + 4·4-s − 7.48·5-s + 13.4·7-s + 8·8-s − 14.9·10-s + 11.9·11-s + 26.9·14-s + 16·16-s − 29.9·20-s + 23.9·22-s + 31.0·25-s + 53.9·28-s − 50·29-s − 61.4·31-s + 32·32-s − 100.·35-s − 59.8·40-s + 47.8·44-s + 132.·49-s + 62.0·50-s − 4.57·53-s − 89.6·55-s + 107.·56-s − 100·58-s + 10·59-s − 122.·62-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 1.49·5-s + 1.92·7-s + 8-s − 1.49·10-s + 1.08·11-s + 1.92·14-s + 16-s − 1.49·20-s + 1.08·22-s + 1.24·25-s + 1.92·28-s − 1.72·29-s − 1.98·31-s + 32-s − 2.88·35-s − 1.49·40-s + 1.08·44-s + 2.71·49-s + 1.24·50-s − 0.0862·53-s − 1.62·55-s + 1.92·56-s − 1.72·58-s + 0.169·59-s − 1.98·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.798346805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.798346805\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 7.48T + 25T^{2} \) |
| 7 | \( 1 - 13.4T + 49T^{2} \) |
| 11 | \( 1 - 11.9T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 50T + 841T^{2} \) |
| 31 | \( 1 + 61.4T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 4.57T + 2.80e3T^{2} \) |
| 59 | \( 1 - 10T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 143.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 58T + 6.24e3T^{2} \) |
| 83 | \( 1 + 17.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 128.T + 9.40e3T^{2} \) |
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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
−11.83407984584653296454658481445, −11.45512249332009800597260272414, −10.81884399482237922896722021657, −8.843526962115495654153450304437, −7.75327755894529546932625397904, −7.20793388823295564740666662322, −5.52627004688533635554540833958, −4.41790692035140023797704053633, −3.71405751954578190198215968930, −1.67024438877746004357248344401,
1.67024438877746004357248344401, 3.71405751954578190198215968930, 4.41790692035140023797704053633, 5.52627004688533635554540833958, 7.20793388823295564740666662322, 7.75327755894529546932625397904, 8.843526962115495654153450304437, 10.81884399482237922896722021657, 11.45512249332009800597260272414, 11.83407984584653296454658481445
