| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 11-s + 12-s + 4·13-s + 14-s + 16-s − 6·17-s − 18-s − 8·19-s − 21-s − 22-s − 6·23-s − 24-s − 4·26-s + 27-s − 28-s − 4·29-s − 2·31-s − 32-s + 33-s + 6·34-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 + 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.218·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s − 0.784·26-s + 0.192·27-s − 0.188·28-s − 0.742·29-s − 0.359·31-s − 0.176·32-s + 0.174·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.350306967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.350306967\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−16.59963319983006, −15.76485734079284, −15.45713994906062, −14.81224024275241, −14.28097047857202, −13.47344240849348, −12.98399398006083, −12.64636249141143, −11.52496724266039, −11.25446011679209, −10.52128620285088, −10.02718911991484, −9.160303718281136, −8.919393878708497, −8.260448831251530, −7.748377064864472, −6.861189771265164, −6.264722631963554, −5.963063301376942, −4.549009401613750, −4.046379719397619, −3.318066432369713, −2.224368978326653, −1.885218952001366, −0.5549374399488318,
0.5549374399488318, 1.885218952001366, 2.224368978326653, 3.318066432369713, 4.046379719397619, 4.549009401613750, 5.963063301376942, 6.264722631963554, 6.861189771265164, 7.748377064864472, 8.260448831251530, 8.919393878708497, 9.160303718281136, 10.02718911991484, 10.52128620285088, 11.25446011679209, 11.52496724266039, 12.64636249141143, 12.98399398006083, 13.47344240849348, 14.28097047857202, 14.81224024275241, 15.45713994906062, 15.76485734079284, 16.59963319983006
