| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 9-s − 11-s − 2·12-s − 13-s − 4·16-s − 17-s − 2·18-s + 2·19-s + 2·22-s + 3·23-s + 2·26-s − 27-s − 2·29-s + 6·31-s + 8·32-s + 33-s + 2·34-s + 2·36-s − 11·37-s − 4·38-s + 39-s + 5·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.277·13-s − 16-s − 0.242·17-s − 0.471·18-s + 0.458·19-s + 0.426·22-s + 0.625·23-s + 0.392·26-s − 0.192·27-s − 0.371·29-s + 1.07·31-s + 1.41·32-s + 0.174·33-s + 0.342·34-s + 1/3·36-s − 1.80·37-s − 0.648·38-s + 0.160·39-s + 0.780·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2168611558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2168611558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.67583894932318, −14.05746502925921, −13.42487592791983, −13.09312493182666, −12.25542391302957, −11.90179359424232, −11.30896640362040, −10.70531154651904, −10.49366005260524, −9.856023563725152, −9.288905015013168, −9.004280269820039, −8.188452072053749, −7.804283956301873, −7.296380576594650, −6.624282855109905, −6.289582275480130, −5.339529144970056, −4.863476234550548, −4.322854481845011, −3.295650413148507, −2.700609386276288, −1.687042453378662, −1.314927515465468, −0.2278307162632441,
0.2278307162632441, 1.314927515465468, 1.687042453378662, 2.700609386276288, 3.295650413148507, 4.322854481845011, 4.863476234550548, 5.339529144970056, 6.289582275480130, 6.624282855109905, 7.296380576594650, 7.804283956301873, 8.188452072053749, 9.004280269820039, 9.288905015013168, 9.856023563725152, 10.49366005260524, 10.70531154651904, 11.30896640362040, 11.90179359424232, 12.25542391302957, 13.09312493182666, 13.42487592791983, 14.05746502925921, 14.67583894932318
