| L(s) = 1 | − 0.0599·2-s + 3-s − 1.99·4-s + 3.32·5-s − 0.0599·6-s + 7-s + 0.239·8-s + 9-s − 0.199·10-s + 1.46·11-s − 1.99·12-s + 2.23·13-s − 0.0599·14-s + 3.32·15-s + 3.97·16-s + 1.17·17-s − 0.0599·18-s − 4.60·19-s − 6.64·20-s + 21-s − 0.0875·22-s + 0.0502·23-s + 0.239·24-s + 6.07·25-s − 0.133·26-s + 27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | − 0.0424·2-s + 0.577·3-s − 0.998·4-s + 1.48·5-s − 0.0244·6-s + 0.377·7-s + 0.0847·8-s + 0.333·9-s − 0.0631·10-s + 0.440·11-s − 0.576·12-s + 0.619·13-s − 0.0160·14-s + 0.859·15-s + 0.994·16-s + 0.285·17-s − 0.0141·18-s − 1.05·19-s − 1.48·20-s + 0.218·21-s − 0.0186·22-s + 0.0104·23-s + 0.0489·24-s + 1.21·25-s − 0.0262·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.846177031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.846177031\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0599T + 2T^{2} \) |
| 5 | \( 1 - 3.32T + 5T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 23 | \( 1 - 0.0502T + 23T^{2} \) |
| 29 | \( 1 + 0.615T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 + 4.46T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 8.08T + 43T^{2} \) |
| 47 | \( 1 + 1.08T + 47T^{2} \) |
| 53 | \( 1 - 8.53T + 53T^{2} \) |
| 59 | \( 1 + 7.69T + 59T^{2} \) |
| 61 | \( 1 - 3.89T + 61T^{2} \) |
| 67 | \( 1 + 1.61T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 6.26T + 73T^{2} \) |
| 79 | \( 1 + 8.30T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 6.63T + 97T^{2} \) |
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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
−8.652722808189197414577735322551, −7.952728011144753035150175475822, −6.98247007007230411175057916464, −5.99551013001511018420509183253, −5.62828071455901053605455815800, −4.55399336382163814667143040402, −3.99482480976025777952742155550, −2.86645368840402419244248358700, −1.91050247465651173178131937378, −1.03444309292827705474770670452,
1.03444309292827705474770670452, 1.91050247465651173178131937378, 2.86645368840402419244248358700, 3.99482480976025777952742155550, 4.55399336382163814667143040402, 5.62828071455901053605455815800, 5.99551013001511018420509183253, 6.98247007007230411175057916464, 7.952728011144753035150175475822, 8.652722808189197414577735322551
