| L(s) = 1 | + 84i·3-s − 456i·7-s − 4.86e3·9-s + 2.52e3·11-s − 1.07e4i·13-s + 1.11e4i·17-s + 4.12e3·19-s + 3.83e4·21-s − 8.17e4i·23-s − 2.25e5i·27-s − 9.97e4·29-s + 4.04e4·31-s + 2.12e5i·33-s + 4.19e5i·37-s + 9.05e5·39-s + ⋯ |
| L(s) = 1 | + 1.79i·3-s − 0.502i·7-s − 2.22·9-s + 0.571·11-s − 1.36i·13-s + 0.550i·17-s + 0.137·19-s + 0.902·21-s − 1.40i·23-s − 2.20i·27-s − 0.759·29-s + 0.244·31-s + 1.02i·33-s + 1.36i·37-s + 2.44·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.841153724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.841153724\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 84iT - 2.18e3T^{2} \) |
| 7 | \( 1 + 456iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 2.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.07e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.11e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 4.12e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.17e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 9.97e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.04e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.19e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 1.41e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.90e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 6.82e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.81e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 9.66e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.88e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.96e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.54e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.68e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 4.03e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.38e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 6.47e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.06e6iT - 8.07e13T^{2} \) |
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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
−10.41085260941103423360285204617, −9.673627988316734660092354038891, −8.741221845375638339307832832095, −7.915202962680560354076133301440, −6.39137175702943625672075632077, −5.39540674156613331853013350269, −4.46425353841342197137473091586, −3.69127342864185342393625339971, −2.74493776405563246547295244909, −0.801130031563294673593534160395,
0.49594915105530604032631254653, 1.64765913618588353887328842344, 2.27363923884452918363110081256, 3.67322154241250741691821710517, 5.31879816168852973316723345619, 6.24228042039588999888915254225, 7.05065493164740279683049047723, 7.68844884530101600791281481349, 8.849985823797735572908367336730, 9.425173843014377251723223847701
