L(s) = 1 | − 5.14·2-s + 6.96·3-s + 18.5·4-s + 3.24·5-s − 35.8·6-s − 54.0·8-s + 21.4·9-s − 16.7·10-s + 58.8·11-s + 128.·12-s + 47.9·13-s + 22.5·15-s + 130.·16-s − 84.9·17-s − 110.·18-s + 54.8·19-s + 60.0·20-s − 302.·22-s + 150.·23-s − 376.·24-s − 114.·25-s − 246.·26-s − 38.6·27-s + 44.2·29-s − 116.·30-s + 95.7·31-s − 238.·32-s + ⋯ |
L(s) = 1 | − 1.82·2-s + 1.33·3-s + 2.31·4-s + 0.290·5-s − 2.43·6-s − 2.38·8-s + 0.794·9-s − 0.528·10-s + 1.61·11-s + 3.09·12-s + 1.02·13-s + 0.389·15-s + 2.03·16-s − 1.21·17-s − 1.44·18-s + 0.662·19-s + 0.671·20-s − 2.93·22-s + 1.36·23-s − 3.19·24-s − 0.915·25-s − 1.86·26-s − 0.275·27-s + 0.283·29-s − 0.708·30-s + 0.554·31-s − 1.31·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.163995407\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.163995407\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + 5.14T + 8T^{2} \) |
| 3 | \( 1 - 6.96T + 27T^{2} \) |
| 5 | \( 1 - 3.24T + 125T^{2} \) |
| 11 | \( 1 - 58.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 47.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 44.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 95.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 209.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 354.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 221.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 508.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 51.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 263.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 203.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 679.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 189.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 520.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 712.T + 9.12e5T^{2} \) |
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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
−8.796553778851834107134702048951, −8.132945926824410541872839338943, −7.42788794486111169904633637033, −6.61061450264803608279713410663, −6.05827338533893803736763863865, −4.30496986035865432272444898907, −3.32757562912131560685982247790, −2.46951904461626751026739785425, −1.59084786794352078216539321658, −0.864584180209969487401598735691,
0.864584180209969487401598735691, 1.59084786794352078216539321658, 2.46951904461626751026739785425, 3.32757562912131560685982247790, 4.30496986035865432272444898907, 6.05827338533893803736763863865, 6.61061450264803608279713410663, 7.42788794486111169904633637033, 8.132945926824410541872839338943, 8.796553778851834107134702048951