| L(s) = 1 | + 2.29·3-s + 1.65·5-s + 7-s + 2.28·9-s + 6.22·11-s − 0.634·13-s + 3.80·15-s − 5.02·17-s + 2.11·19-s + 2.29·21-s + 8.89·23-s − 2.26·25-s − 1.63·27-s − 1.13·29-s + 8.27·31-s + 14.3·33-s + 1.65·35-s − 2.19·37-s − 1.45·39-s − 4.93·41-s − 5.27·43-s + 3.78·45-s + 6.68·47-s + 49-s − 11.5·51-s + 6.41·53-s + 10.2·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 0.739·5-s + 0.377·7-s + 0.762·9-s + 1.87·11-s − 0.175·13-s + 0.981·15-s − 1.21·17-s + 0.486·19-s + 0.501·21-s + 1.85·23-s − 0.453·25-s − 0.315·27-s − 0.210·29-s + 1.48·31-s + 2.49·33-s + 0.279·35-s − 0.360·37-s − 0.233·39-s − 0.769·41-s − 0.805·43-s + 0.563·45-s + 0.975·47-s + 0.142·49-s − 1.61·51-s + 0.880·53-s + 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.743840793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.743840793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 - 1.65T + 5T^{2} \) |
| 11 | \( 1 - 6.22T + 11T^{2} \) |
| 13 | \( 1 + 0.634T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 - 8.89T + 23T^{2} \) |
| 29 | \( 1 + 1.13T + 29T^{2} \) |
| 31 | \( 1 - 8.27T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 - 6.41T + 53T^{2} \) |
| 59 | \( 1 + 1.50T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 4.07T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 9.19T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 - 1.60T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.167437096542751558166448575829, −7.08307169935445627082499307002, −6.79042285867851500492872913498, −5.92007494669855446436204354171, −4.95925240634741709123167993890, −4.21413572817218927004961418868, −3.47654292090229732565215327529, −2.66650969943729044098254210918, −1.90017172662275584668922331137, −1.13661785374164485207396009259,
1.13661785374164485207396009259, 1.90017172662275584668922331137, 2.66650969943729044098254210918, 3.47654292090229732565215327529, 4.21413572817218927004961418868, 4.95925240634741709123167993890, 5.92007494669855446436204354171, 6.79042285867851500492872913498, 7.08307169935445627082499307002, 8.167437096542751558166448575829
