| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s + 6·13-s − 15-s − 4·19-s − 8·23-s + 25-s − 27-s + 6·29-s + 4·31-s + 4·33-s − 2·37-s − 6·39-s − 6·41-s + 4·43-s + 45-s − 4·47-s − 7·49-s + 10·53-s − 4·55-s + 4·57-s − 12·59-s − 6·61-s + 6·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.258·15-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.328·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s − 49-s + 1.37·53-s − 0.539·55-s + 0.529·57-s − 1.56·59-s − 0.768·61-s + 0.744·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.267416596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267416596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.99080584900570, −13.65074548645903, −13.04253222031639, −12.85731663887896, −11.98442917951755, −11.78711115946839, −10.96387677603017, −10.62079494269614, −10.21463752500358, −9.823820060905659, −8.975077399595546, −8.317122054565853, −8.249096432406123, −7.435682678320041, −6.714679114718276, −6.123417084339866, −5.977068625558145, −5.288109146698472, −4.575768327038694, −4.173021535118584, −3.342962788976036, −2.734072227865354, −1.916838154163154, −1.374999391891432, −0.3897631740076203,
0.3897631740076203, 1.374999391891432, 1.916838154163154, 2.734072227865354, 3.342962788976036, 4.173021535118584, 4.575768327038694, 5.288109146698472, 5.977068625558145, 6.123417084339866, 6.714679114718276, 7.435682678320041, 8.249096432406123, 8.317122054565853, 8.975077399595546, 9.823820060905659, 10.21463752500358, 10.62079494269614, 10.96387677603017, 11.78711115946839, 11.98442917951755, 12.85731663887896, 13.04253222031639, 13.65074548645903, 13.99080584900570
