| L(s) = 1 | + 3-s + 5-s + 9-s − 4·11-s − 6·13-s + 15-s + 6·17-s + 4·19-s + 25-s + 27-s + 2·29-s − 4·33-s + 2·37-s − 6·39-s − 6·41-s + 12·43-s + 45-s − 8·47-s − 7·49-s + 6·51-s − 6·53-s − 4·55-s + 4·57-s − 12·59-s − 14·61-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 + 1.45·17-s + 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.696·33-s + 0.328·37-s − 0.960·39-s − 0.937·41-s + 1.82·43-s + 0.149·45-s − 1.16·47-s − 49-s + 0.840·51-s − 0.824·53-s − 0.539·55-s + 0.529·57-s − 1.56·59-s − 1.79·61-s − 0.744·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.824063225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.824063225\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.86961089248663, −12.56002236741189, −12.07735697920975, −11.69991709096753, −10.92213938240516, −10.47894670478632, −10.03998237570151, −9.612619664306420, −9.427121109315697, −8.723830111598900, −7.997274889644867, −7.724306159655126, −7.480137062478980, −6.862618288047183, −6.163194001379347, −5.626149089079437, −5.153155874271284, −4.769609660128415, −4.231704027394262, −3.268355681073516, −2.859832221841295, −2.727536018170472, −1.745695145852635, −1.364065765240687, −0.3418510599653285,
0.3418510599653285, 1.364065765240687, 1.745695145852635, 2.727536018170472, 2.859832221841295, 3.268355681073516, 4.231704027394262, 4.769609660128415, 5.153155874271284, 5.626149089079437, 6.163194001379347, 6.862618288047183, 7.480137062478980, 7.724306159655126, 7.997274889644867, 8.723830111598900, 9.427121109315697, 9.612619664306420, 10.03998237570151, 10.47894670478632, 10.92213938240516, 11.69991709096753, 12.07735697920975, 12.56002236741189, 12.86961089248663
