| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 8-s + 9-s + 2·10-s + 2·11-s − 12-s − 4·13-s + 2·15-s + 16-s − 18-s + 2·19-s − 2·20-s − 2·22-s + 4·23-s + 24-s − 25-s + 4·26-s − 27-s − 2·30-s − 32-s − 2·33-s + 36-s + 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.603·11-s − 0.288·12-s − 1.10·13-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.447·20-s − 0.426·22-s + 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.365·30-s − 0.176·32-s − 0.348·33-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7922109170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7922109170\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.11123816509433, −13.23020370154042, −12.83517857734246, −12.21924447307860, −11.81779874026984, −11.48268937725579, −11.06081200979270, −10.44971424399884, −9.846444702597976, −9.524546949185572, −8.959928938081281, −8.324941378420162, −7.748679912650427, −7.423817947634496, −6.863897059262733, −6.403260108031613, −5.731688351137570, −5.042874149490582, −4.601119605699861, −3.909791878294989, −3.304095921602136, −2.622075954993829, −1.849614851425886, −1.037495399801443, −0.3952232037773752,
0.3952232037773752, 1.037495399801443, 1.849614851425886, 2.622075954993829, 3.304095921602136, 3.909791878294989, 4.601119605699861, 5.042874149490582, 5.731688351137570, 6.403260108031613, 6.863897059262733, 7.423817947634496, 7.748679912650427, 8.324941378420162, 8.959928938081281, 9.524546949185572, 9.846444702597976, 10.44971424399884, 11.06081200979270, 11.48268937725579, 11.81779874026984, 12.21924447307860, 12.83517857734246, 13.23020370154042, 14.11123816509433
