L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 4·11-s − 12-s + 2·13-s − 14-s + 16-s + 18-s + 21-s + 4·22-s − 8·23-s − 24-s + 2·26-s − 27-s − 28-s − 10·29-s + 8·31-s + 32-s − 4·33-s + 36-s + 2·37-s − 2·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.218·21-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + 1/6·36-s + 0.328·37-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.110437434\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.110437434\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.67205866818347, −12.16776376832543, −11.74183048487389, −11.46952816561048, −11.07240842687450, −10.42163627388649, −9.974502156040074, −9.570478112639627, −9.126520311326275, −8.426826230198364, −7.969405149391916, −7.502274527110926, −6.796680091397685, −6.452214753222432, −6.107456180908263, −5.730993523440860, −5.020887265561644, −4.592889429993563, −3.927369802457071, −3.675350871955015, −3.193890162780535, −2.229514954322600, −1.850019660012788, −1.173632279657419, −0.4433681300794182,
0.4433681300794182, 1.173632279657419, 1.850019660012788, 2.229514954322600, 3.193890162780535, 3.675350871955015, 3.927369802457071, 4.592889429993563, 5.020887265561644, 5.730993523440860, 6.107456180908263, 6.452214753222432, 6.796680091397685, 7.502274527110926, 7.969405149391916, 8.426826230198364, 9.126520311326275, 9.570478112639627, 9.974502156040074, 10.42163627388649, 11.07240842687450, 11.46952816561048, 11.74183048487389, 12.16776376832543, 12.67205866818347