| L(s) = 1 | − 2.60·2-s + 3·3-s − 1.23·4-s − 11.0·5-s − 7.80·6-s − 16.8·7-s + 24.0·8-s + 9·9-s + 28.7·10-s − 0.0148·11-s − 3.71·12-s − 3.98·13-s + 43.8·14-s − 33.1·15-s − 52.5·16-s + 111.·17-s − 23.4·18-s − 51.9·19-s + 13.7·20-s − 50.5·21-s + 0.0384·22-s + 134.·23-s + 72.0·24-s − 2.59·25-s + 10.3·26-s + 27·27-s + 20.8·28-s + ⋯ |
| L(s) = 1 | − 0.919·2-s + 0.577·3-s − 0.154·4-s − 0.989·5-s − 0.530·6-s − 0.910·7-s + 1.06·8-s + 0.333·9-s + 0.909·10-s − 0.000405·11-s − 0.0893·12-s − 0.0849·13-s + 0.836·14-s − 0.571·15-s − 0.821·16-s + 1.58·17-s − 0.306·18-s − 0.627·19-s + 0.153·20-s − 0.525·21-s + 0.000373·22-s + 1.21·23-s + 0.612·24-s − 0.0207·25-s + 0.0780·26-s + 0.192·27-s + 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 717 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 2.60T + 8T^{2} \) |
| 5 | \( 1 + 11.0T + 125T^{2} \) |
| 7 | \( 1 + 16.8T + 343T^{2} \) |
| 11 | \( 1 + 0.0148T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.98T + 2.19e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 391.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 345.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 442.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 222.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 665.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 584.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 21.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 427.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 66.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 342.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 884.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 703.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 960.T + 9.12e5T^{2} \) |
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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
−9.536064371208422660995972377980, −8.763339374653553016304505644125, −7.86051018830252691525541199344, −7.48263184466374088655767784794, −6.29199824930049343802107139667, −4.81862191027833188115355661676, −3.82722692312882309574972828758, −2.92181690337253350570376900153, −1.18105481048354542095854644612, 0,
1.18105481048354542095854644612, 2.92181690337253350570376900153, 3.82722692312882309574972828758, 4.81862191027833188115355661676, 6.29199824930049343802107139667, 7.48263184466374088655767784794, 7.86051018830252691525541199344, 8.763339374653553016304505644125, 9.536064371208422660995972377980
