| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 13-s − 14-s − 15-s + 16-s − 18-s − 6·19-s + 20-s − 21-s − 22-s + 3·23-s + 24-s + 25-s + 26-s − 27-s + 28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.218·21-s − 0.213·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4498020415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4498020415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.59203697278506, −11.73261576014151, −11.57477782343352, −11.14822268753650, −10.61868580531505, −10.29137903282962, −9.785922308395002, −9.393500382166717, −8.818718270591799, −8.498076595292413, −7.998459224083389, −7.287660057515402, −7.111799426609120, −6.406818507210201, −6.154517735418943, −5.604665057492656, −4.983835224142816, −4.638088228879818, −4.019723889934139, −3.327234043669532, −2.813447912896134, −1.937560128662813, −1.770630300053183, −1.111433450191883, −0.2075280315841309,
0.2075280315841309, 1.111433450191883, 1.770630300053183, 1.937560128662813, 2.813447912896134, 3.327234043669532, 4.019723889934139, 4.638088228879818, 4.983835224142816, 5.604665057492656, 6.154517735418943, 6.406818507210201, 7.111799426609120, 7.287660057515402, 7.998459224083389, 8.498076595292413, 8.818718270591799, 9.393500382166717, 9.785922308395002, 10.29137903282962, 10.61868580531505, 11.14822268753650, 11.57477782343352, 11.73261576014151, 12.59203697278506
