| L(s) = 1 | − 2.23·2-s + 3-s + 3.01·4-s + 2.46·5-s − 2.23·6-s − 2.26·8-s + 9-s − 5.51·10-s + 3.07·11-s + 3.01·12-s + 3.08·13-s + 2.46·15-s − 0.953·16-s − 3.92·17-s − 2.23·18-s + 2.23·19-s + 7.41·20-s − 6.87·22-s − 23-s − 2.26·24-s + 1.06·25-s − 6.89·26-s + 27-s + 4.31·29-s − 5.51·30-s + 3.47·31-s + 6.66·32-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.50·4-s + 1.10·5-s − 0.913·6-s − 0.800·8-s + 0.333·9-s − 1.74·10-s + 0.926·11-s + 0.869·12-s + 0.854·13-s + 0.635·15-s − 0.238·16-s − 0.952·17-s − 0.527·18-s + 0.513·19-s + 1.65·20-s − 1.46·22-s − 0.208·23-s − 0.462·24-s + 0.213·25-s − 1.35·26-s + 0.192·27-s + 0.801·29-s − 1.00·30-s + 0.624·31-s + 1.17·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.517829020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.517829020\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 11 | \( 1 - 3.07T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 + 3.92T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 29 | \( 1 - 4.31T + 29T^{2} \) |
| 31 | \( 1 - 3.47T + 31T^{2} \) |
| 37 | \( 1 + 3.10T + 37T^{2} \) |
| 41 | \( 1 - 8.45T + 41T^{2} \) |
| 43 | \( 1 + 3.01T + 43T^{2} \) |
| 47 | \( 1 + 9.63T + 47T^{2} \) |
| 53 | \( 1 - 3.32T + 53T^{2} \) |
| 59 | \( 1 - 7.60T + 59T^{2} \) |
| 61 | \( 1 + 1.60T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 + 9.72T + 79T^{2} \) |
| 83 | \( 1 + 2.34T + 83T^{2} \) |
| 89 | \( 1 - 5.01T + 89T^{2} \) |
| 97 | \( 1 - 5.86T + 97T^{2} \) |
| 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
−8.622647126973249106993252421829, −8.293768520607877637055895886127, −7.24281437024556129780331429626, −6.57085227603272450701759044244, −6.04230073005963028584018195384, −4.78445344362906001750551224043, −3.72150627842567492507793809179, −2.53719037585441741855944853521, −1.78401650124264498128028464061, −0.968986014427637494419018884979,
0.968986014427637494419018884979, 1.78401650124264498128028464061, 2.53719037585441741855944853521, 3.72150627842567492507793809179, 4.78445344362906001750551224043, 6.04230073005963028584018195384, 6.57085227603272450701759044244, 7.24281437024556129780331429626, 8.293768520607877637055895886127, 8.622647126973249106993252421829
