| L(s) = 1 | + 2-s − 0.618·3-s + 4-s + 5-s − 0.618·6-s + 1.61·7-s + 8-s − 2.61·9-s + 10-s + 3.85·11-s − 0.618·12-s + 4.09·13-s + 1.61·14-s − 0.618·15-s + 16-s − 5.09·17-s − 2.61·18-s − 4.85·19-s + 20-s − 1.00·21-s + 3.85·22-s + 23-s − 0.618·24-s + 25-s + 4.09·26-s + 3.47·27-s + 1.61·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.356·3-s + 0.5·4-s + 0.447·5-s − 0.252·6-s + 0.611·7-s + 0.353·8-s − 0.872·9-s + 0.316·10-s + 1.16·11-s − 0.178·12-s + 1.13·13-s + 0.432·14-s − 0.159·15-s + 0.250·16-s − 1.23·17-s − 0.617·18-s − 1.11·19-s + 0.223·20-s − 0.218·21-s + 0.821·22-s + 0.208·23-s − 0.126·24-s + 0.200·25-s + 0.802·26-s + 0.668·27-s + 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801075793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801075793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 3.85T + 11T^{2} \) |
| 13 | \( 1 - 4.09T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 9.70T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 - 7.09T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−12.15772750434465927058119918655, −11.19558779221915460508966203868, −10.82084895697821165088256119833, −9.140306759551471116968745802587, −8.377340443911295124746456879666, −6.70407123397807318145608994971, −6.06854457591781683225897468344, −4.89069048655957089784278056400, −3.67058546540356043216825506916, −1.89317783414266679508333114377,
1.89317783414266679508333114377, 3.67058546540356043216825506916, 4.89069048655957089784278056400, 6.06854457591781683225897468344, 6.70407123397807318145608994971, 8.377340443911295124746456879666, 9.140306759551471116968745802587, 10.82084895697821165088256119833, 11.19558779221915460508966203868, 12.15772750434465927058119918655
