| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s − 13-s + 14-s − 15-s + 16-s − 2·17-s − 18-s − 19-s + 20-s + 21-s + 4·22-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 − 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.06237211768733, −14.68701321587419, −13.83762490967546, −13.21762970579743, −13.11525151354102, −12.21211434905706, −12.05868640472437, −11.15974642797959, −10.82292362230593, −10.29636539260288, −9.963002106488847, −9.324861744849884, −8.797151530150690, −8.255752361690366, −7.556661629449552, −7.117088118630656, −6.575124813279129, −5.956870398043441, −5.419597713539800, −4.941167603927698, −4.179992797505563, −3.283113969742184, −2.715790144682215, −1.956103226815145, −1.372888772519076, 0, 0,
1.372888772519076, 1.956103226815145, 2.715790144682215, 3.283113969742184, 4.179992797505563, 4.941167603927698, 5.419597713539800, 5.956870398043441, 6.575124813279129, 7.117088118630656, 7.556661629449552, 8.255752361690366, 8.797151530150690, 9.324861744849884, 9.963002106488847, 10.29636539260288, 10.82292362230593, 11.15974642797959, 12.05868640472437, 12.21211434905706, 13.11525151354102, 13.21762970579743, 13.83762490967546, 14.68701321587419, 15.06237211768733
