| L(s) = 1 | − 2-s + 3-s + 4-s − 4·5-s − 6-s + 3·7-s − 8-s − 2·9-s + 4·10-s + 2·11-s + 12-s + 13-s − 3·14-s − 4·15-s + 16-s + 3·17-s + 2·18-s − 4·20-s + 3·21-s − 2·22-s − 23-s − 24-s + 11·25-s − 26-s − 5·27-s + 3·28-s + 5·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s − 2/3·9-s + 1.26·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s − 0.801·14-s − 1.03·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.894·20-s + 0.654·21-s − 0.426·22-s − 0.208·23-s − 0.204·24-s + 11/5·25-s − 0.196·26-s − 0.962·27-s + 0.566·28-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080989695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080989695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.55928337618867689245360210141, −9.228932927125891932739757326156, −8.474637821196586665001996068247, −7.957421913838810837694491246300, −7.46242727862265190447093907017, −6.16947011382354679761132974999, −4.71396131109660693238138315060, −3.78873474603954853324187046999, −2.72199195247203046939301106692, −0.980313161146313482644993247523,
0.980313161146313482644993247523, 2.72199195247203046939301106692, 3.78873474603954853324187046999, 4.71396131109660693238138315060, 6.16947011382354679761132974999, 7.46242727862265190447093907017, 7.957421913838810837694491246300, 8.474637821196586665001996068247, 9.228932927125891932739757326156, 10.55928337618867689245360210141
