| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 11·5-s − 6·6-s + 19·7-s − 8·8-s − 18·9-s − 22·10-s − 38·11-s + 12·12-s − 13·13-s − 38·14-s + 33·15-s + 16·16-s − 51·17-s + 36·18-s + 90·19-s + 44·20-s + 57·21-s + 76·22-s − 52·23-s − 24·24-s − 4·25-s + 26·26-s − 135·27-s + 76·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.983·5-s − 0.408·6-s + 1.02·7-s − 0.353·8-s − 2/3·9-s − 0.695·10-s − 1.04·11-s + 0.288·12-s − 0.277·13-s − 0.725·14-s + 0.568·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 1.08·19-s + 0.491·20-s + 0.592·21-s + 0.736·22-s − 0.471·23-s − 0.204·24-s − 0.0319·25-s + 0.196·26-s − 0.962·27-s + 0.512·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.119117757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.119117757\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 3 | \( 1 - p T + p^{3} T^{2} \) |
| 5 | \( 1 - 11 T + p^{3} T^{2} \) |
| 7 | \( 1 - 19 T + p^{3} T^{2} \) |
| 11 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 90 T + p^{3} T^{2} \) |
| 23 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 292 T + p^{3} T^{2} \) |
| 37 | \( 1 + 441 T + p^{3} T^{2} \) |
| 41 | \( 1 - 312 T + p^{3} T^{2} \) |
| 43 | \( 1 - 373 T + p^{3} T^{2} \) |
| 47 | \( 1 + 41 T + p^{3} T^{2} \) |
| 53 | \( 1 - 468 T + p^{3} T^{2} \) |
| 59 | \( 1 - 530 T + p^{3} T^{2} \) |
| 61 | \( 1 - 592 T + p^{3} T^{2} \) |
| 67 | \( 1 + 206 T + p^{3} T^{2} \) |
| 71 | \( 1 + 863 T + p^{3} T^{2} \) |
| 73 | \( 1 + 322 T + p^{3} T^{2} \) |
| 79 | \( 1 + 460 T + p^{3} T^{2} \) |
| 83 | \( 1 - 528 T + p^{3} T^{2} \) |
| 89 | \( 1 - 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 346 T + p^{3} T^{2} \) |
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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
−17.44897771017306689358530248004, −15.79101133149418082435901074445, −14.45347470891581209499784903712, −13.48182305516904666332503457499, −11.56846251749008817090355118332, −10.20943786841303125251742181304, −8.882665630376888335422700096578, −7.68213431936305628037159134047, −5.53357861054215551559310159046, −2.30891282118984490867625239305,
2.30891282118984490867625239305, 5.53357861054215551559310159046, 7.68213431936305628037159134047, 8.882665630376888335422700096578, 10.20943786841303125251742181304, 11.56846251749008817090355118332, 13.48182305516904666332503457499, 14.45347470891581209499784903712, 15.79101133149418082435901074445, 17.44897771017306689358530248004
