| L(s) = 1 | + 3-s + 9-s − 3·11-s + 4·13-s + 4·19-s − 8·23-s + 27-s + 3·29-s − 5·31-s − 3·33-s + 8·37-s + 4·39-s + 8·41-s + 6·43-s − 10·47-s + 9·53-s + 4·57-s + 5·59-s + 10·61-s + 6·67-s − 8·69-s + 10·71-s − 2·73-s + 11·79-s + 81-s + 7·83-s + 3·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.917·19-s − 1.66·23-s + 0.192·27-s + 0.557·29-s − 0.898·31-s − 0.522·33-s + 1.31·37-s + 0.640·39-s + 1.24·41-s + 0.914·43-s − 1.45·47-s + 1.23·53-s + 0.529·57-s + 0.650·59-s + 1.28·61-s + 0.733·67-s − 0.963·69-s + 1.18·71-s − 0.234·73-s + 1.23·79-s + 1/9·81-s + 0.768·83-s + 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.884951328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.884951328\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.13706295934085, −12.48342105385104, −12.07955004369125, −11.35522988018363, −11.14375110881592, −10.58826388311087, −9.943941446143974, −9.753402646414710, −9.201257860743890, −8.543260892265012, −8.245967240921573, −7.734436469504895, −7.448440191214174, −6.708200131033864, −6.198886676317642, −5.687974699339317, −5.283202604638216, −4.582027555515558, −3.937898467892491, −3.672297479120298, −2.986257395916083, −2.358346739541912, −1.997597647678537, −1.078412726170906, −0.5788378519830668,
0.5788378519830668, 1.078412726170906, 1.997597647678537, 2.358346739541912, 2.986257395916083, 3.672297479120298, 3.937898467892491, 4.582027555515558, 5.283202604638216, 5.687974699339317, 6.198886676317642, 6.708200131033864, 7.448440191214174, 7.734436469504895, 8.245967240921573, 8.543260892265012, 9.201257860743890, 9.753402646414710, 9.943941446143974, 10.58826388311087, 11.14375110881592, 11.35522988018363, 12.07955004369125, 12.48342105385104, 13.13706295934085
