| L(s) = 1 | + 2-s − 2.29·3-s + 4-s − 3.07·5-s − 2.29·6-s + 7-s + 8-s + 2.26·9-s − 3.07·10-s − 2.58·11-s − 2.29·12-s + 3.65·13-s + 14-s + 7.05·15-s + 16-s − 7.38·17-s + 2.26·18-s + 0.834·19-s − 3.07·20-s − 2.29·21-s − 2.58·22-s + 4.77·23-s − 2.29·24-s + 4.44·25-s + 3.65·26-s + 1.67·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.32·3-s + 0.5·4-s − 1.37·5-s − 0.937·6-s + 0.377·7-s + 0.353·8-s + 0.756·9-s − 0.971·10-s − 0.780·11-s − 0.662·12-s + 1.01·13-s + 0.267·14-s + 1.82·15-s + 0.250·16-s − 1.79·17-s + 0.534·18-s + 0.191·19-s − 0.687·20-s − 0.500·21-s − 0.551·22-s + 0.995·23-s − 0.468·24-s + 0.888·25-s + 0.716·26-s + 0.322·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109511084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109511084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 + 3.07T + 5T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 - 3.65T + 13T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 - 0.834T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 - 6.15T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 1.04T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 9.41T + 71T^{2} \) |
| 73 | \( 1 + 8.07T + 73T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 - 2.81T + 89T^{2} \) |
| 97 | \( 1 - 4.45T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.39291263471601926387414234151, −8.806884028590757743983579305311, −8.069713038846132105860309165555, −7.08201428186905458983565709645, −6.42510688954777058545905434547, −5.45033563715303447038523063598, −4.63173388520060837279686912185, −4.03375038826022054288900723380, −2.69377906661848549324473330955, −0.76452042604513579335813713169,
0.76452042604513579335813713169, 2.69377906661848549324473330955, 4.03375038826022054288900723380, 4.63173388520060837279686912185, 5.45033563715303447038523063598, 6.42510688954777058545905434547, 7.08201428186905458983565709645, 8.069713038846132105860309165555, 8.806884028590757743983579305311, 10.39291263471601926387414234151
