| L(s) = 1 | − 1.78·2-s + 3-s + 1.19·4-s − 3.42·5-s − 1.78·6-s + 1.44·8-s + 9-s + 6.12·10-s + 1.59·11-s + 1.19·12-s + 13-s − 3.42·15-s − 4.96·16-s − 0.0394·17-s − 1.78·18-s + 3.64·19-s − 4.08·20-s − 2.84·22-s − 3.42·23-s + 1.44·24-s + 6.76·25-s − 1.78·26-s + 27-s − 10.0·29-s + 6.12·30-s + 6.15·31-s + 5.97·32-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.577·3-s + 0.595·4-s − 1.53·5-s − 0.729·6-s + 0.510·8-s + 0.333·9-s + 1.93·10-s + 0.480·11-s + 0.343·12-s + 0.277·13-s − 0.885·15-s − 1.24·16-s − 0.00957·17-s − 0.421·18-s + 0.835·19-s − 0.913·20-s − 0.607·22-s − 0.713·23-s + 0.294·24-s + 1.35·25-s − 0.350·26-s + 0.192·27-s − 1.85·29-s + 1.11·30-s + 1.10·31-s + 1.05·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6922583436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6922583436\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.78T + 2T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 17 | \( 1 + 0.0394T + 17T^{2} \) |
| 19 | \( 1 - 3.64T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 - 6.15T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.19T + 59T^{2} \) |
| 61 | \( 1 - 1.96T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 1.79T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 5.34T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
| 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
−9.120069328899201004351300466409, −8.418552802594076436679692916415, −7.72117649870513490472817927844, −7.43892353423307948324619140174, −6.43873029443677427848095876891, −4.97818252585471738876309417596, −4.00088367559596072844068591007, −3.40603050020536800684714228040, −1.92521511251875494213025229481, −0.66543060780063271262703265358,
0.66543060780063271262703265358, 1.92521511251875494213025229481, 3.40603050020536800684714228040, 4.00088367559596072844068591007, 4.97818252585471738876309417596, 6.43873029443677427848095876891, 7.43892353423307948324619140174, 7.72117649870513490472817927844, 8.418552802594076436679692916415, 9.120069328899201004351300466409
