| L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s − 3·8-s + 9-s − 2·10-s + 11-s − 12-s − 2·15-s − 16-s − 2·17-s + 18-s + 4·19-s + 2·20-s + 22-s − 3·24-s − 25-s + 27-s − 2·29-s − 2·30-s + 8·31-s + 5·32-s + 33-s − 2·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s − 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.213·22-s − 0.612·24-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.883·32-s + 0.174·33-s − 0.342·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.315677501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.315677501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.89632097125228, −12.32048365476283, −11.83009097629736, −11.70574015520308, −11.06565910456552, −10.49616526988381, −9.799618134779705, −9.558935008094289, −9.078196377504634, −8.413005309063417, −8.248318404468871, −7.705795004383067, −7.138509172596178, −6.604709444701420, −6.175936560612405, −5.413461113386694, −5.037683208308491, −4.515521038212013, −3.977696050335610, −3.612072364580318, −3.236146627217582, −2.523768899499871, −1.985444563073240, −0.9563644894770993, −0.5108526991881828,
0.5108526991881828, 0.9563644894770993, 1.985444563073240, 2.523768899499871, 3.236146627217582, 3.612072364580318, 3.977696050335610, 4.515521038212013, 5.037683208308491, 5.413461113386694, 6.175936560612405, 6.604709444701420, 7.138509172596178, 7.705795004383067, 8.248318404468871, 8.413005309063417, 9.078196377504634, 9.558935008094289, 9.799618134779705, 10.49616526988381, 11.06565910456552, 11.70574015520308, 11.83009097629736, 12.32048365476283, 12.89632097125228
