| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s + 11-s − 12-s − 13-s + 14-s + 2·15-s + 16-s + 17-s − 18-s + 8·19-s − 2·20-s + 21-s − 22-s + 2·23-s + 24-s − 25-s + 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.447·20-s + 0.218·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.31731170588816, −15.93683994675943, −15.53047272949540, −14.89589600448706, −14.18344494689622, −13.62408986607141, −12.71949206068455, −12.30829994766365, −11.72181296294013, −11.37031809048926, −10.73560684710777, −10.08844065709744, −9.472643926713480, −9.066118962170226, −8.192371282847917, −7.569642023673164, −7.195858440766181, −6.587331630762856, −5.697413400963170, −5.225877937191784, −4.311434694393369, −3.507100120566916, −3.001727381688874, −1.780338488639793, −0.9028062750236141, 0,
0.9028062750236141, 1.780338488639793, 3.001727381688874, 3.507100120566916, 4.311434694393369, 5.225877937191784, 5.697413400963170, 6.587331630762856, 7.195858440766181, 7.569642023673164, 8.192371282847917, 9.066118962170226, 9.472643926713480, 10.08844065709744, 10.73560684710777, 11.37031809048926, 11.72181296294013, 12.30829994766365, 12.71949206068455, 13.62408986607141, 14.18344494689622, 14.89589600448706, 15.53047272949540, 15.93683994675943, 16.31731170588816
