| 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 − 2·13-s + 2·15-s − 16-s − 18-s − 4·19-s + 2·20-s + 22-s − 23-s − 3·24-s − 25-s + 2·26-s − 27-s − 2·29-s − 2·30-s + 8·31-s − 5·32-s + 33-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 − 0.554·13-s + 0.516·15-s − 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.213·22-s − 0.208·23-s − 0.612·24-s − 1/5·25-s + 0.392·26-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37191 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−15.39858596192688, −15.07888301753988, −14.40175744107809, −13.68748105451428, −13.36208125840259, −12.65149417427711, −12.15627131613527, −11.77529492575983, −11.01235733823095, −10.69562870663636, −9.942473293674098, −9.786307680244519, −8.898768982501944, −8.388505664967680, −7.960233609224945, −7.466931348974591, −6.794119422149263, −6.275047482973480, −5.297445891749272, −4.949246172258452, −4.228228202765497, −3.846093647566169, −2.919452296896679, −1.966395948140900, −1.177066076739011, 0, 0,
1.177066076739011, 1.966395948140900, 2.919452296896679, 3.846093647566169, 4.228228202765497, 4.949246172258452, 5.297445891749272, 6.275047482973480, 6.794119422149263, 7.466931348974591, 7.960233609224945, 8.388505664967680, 8.898768982501944, 9.786307680244519, 9.942473293674098, 10.69562870663636, 11.01235733823095, 11.77529492575983, 12.15627131613527, 12.65149417427711, 13.36208125840259, 13.68748105451428, 14.40175744107809, 15.07888301753988, 15.39858596192688
