| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 16-s + 25-s − 27-s + 36-s − 48-s + 49-s − 2·61-s + 64-s − 2·67-s − 75-s − 2·79-s + 81-s − 2·97-s + 100-s − 108-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 16-s + 25-s − 27-s + 36-s − 48-s + 49-s − 2·61-s + 64-s − 2·67-s − 75-s − 2·79-s + 81-s − 2·97-s + 100-s − 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9067870278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9067870278\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 241 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−10.69614521949851227294624788069, −10.12378663947451621928447091099, −8.971583387001835703377812204928, −7.71524261488400082238368692332, −7.00082847724735716671465788696, −6.21272073372215466291200565772, −5.42137648164667980200530959935, −4.30002204280268940541511136326, −2.90974091385881783754263237073, −1.46230540906363762057050556706,
1.46230540906363762057050556706, 2.90974091385881783754263237073, 4.30002204280268940541511136326, 5.42137648164667980200530959935, 6.21272073372215466291200565772, 7.00082847724735716671465788696, 7.71524261488400082238368692332, 8.971583387001835703377812204928, 10.12378663947451621928447091099, 10.69614521949851227294624788069
