| L(s) = 1 | + 2-s − 3-s + 4-s − 0.592·5-s − 6-s + 4.36·7-s + 8-s + 9-s − 0.592·10-s − 0.525·11-s − 12-s − 13-s + 4.36·14-s + 0.592·15-s + 16-s − 1.78·17-s + 18-s + 2.41·19-s − 0.592·20-s − 4.36·21-s − 0.525·22-s − 6.72·23-s − 24-s − 4.64·25-s − 26-s − 27-s + 4.36·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.265·5-s − 0.408·6-s + 1.64·7-s + 0.353·8-s + 0.333·9-s − 0.187·10-s − 0.158·11-s − 0.288·12-s − 0.277·13-s + 1.16·14-s + 0.153·15-s + 0.250·16-s − 0.431·17-s + 0.235·18-s + 0.553·19-s − 0.132·20-s − 0.951·21-s − 0.112·22-s − 1.40·23-s − 0.204·24-s − 0.929·25-s − 0.196·26-s − 0.192·27-s + 0.824·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.078153778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.078153778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 5 | \( 1 + 0.592T + 5T^{2} \) |
| 7 | \( 1 - 4.36T + 7T^{2} \) |
| 11 | \( 1 + 0.525T + 11T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 - 2.41T + 19T^{2} \) |
| 23 | \( 1 + 6.72T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 - 7.98T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 - 2.75T + 43T^{2} \) |
| 47 | \( 1 + 0.804T + 47T^{2} \) |
| 53 | \( 1 + 6.78T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 + 0.222T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 5.16T + 73T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 9.30T + 89T^{2} \) |
| 97 | \( 1 - 0.545T + 97T^{2} \) |
| 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
−7.966137149035018977964501060014, −7.41923033662070591158802583868, −6.45245460173231848997827855024, −5.81672656681385341112304295909, −5.11102133274434999497173192732, −4.42080719513011853928128436147, −4.06634053415283234365054584970, −2.69874711389970953376349690555, −1.92800345114224792279634491402, −0.877074602198305458514802214730,
0.877074602198305458514802214730, 1.92800345114224792279634491402, 2.69874711389970953376349690555, 4.06634053415283234365054584970, 4.42080719513011853928128436147, 5.11102133274434999497173192732, 5.81672656681385341112304295909, 6.45245460173231848997827855024, 7.41923033662070591158802583868, 7.966137149035018977964501060014
