L(s) = 1 | − 3·3-s − 4.54·5-s − 7·7-s + 9·9-s + 40.7·11-s + 53.2·13-s + 13.6·15-s + 4.54·17-s − 122.·19-s + 21·21-s − 131.·23-s − 104.·25-s − 27·27-s − 216.·29-s + 251.·31-s − 122.·33-s + 31.8·35-s + 11.8·37-s − 159.·39-s − 111.·41-s − 369.·43-s − 40.9·45-s + 262.·47-s + 49·49-s − 13.6·51-s − 567.·53-s − 185.·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.406·5-s − 0.377·7-s + 0.333·9-s + 1.11·11-s + 1.13·13-s + 0.234·15-s + 0.0649·17-s − 1.48·19-s + 0.218·21-s − 1.19·23-s − 0.834·25-s − 0.192·27-s − 1.38·29-s + 1.45·31-s − 0.644·33-s + 0.153·35-s + 0.0528·37-s − 0.656·39-s − 0.425·41-s − 1.30·43-s − 0.135·45-s + 0.815·47-s + 0.142·49-s − 0.0374·51-s − 1.46·53-s − 0.454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
good | 5 | \( 1 + 4.54T + 125T^{2} \) |
| 11 | \( 1 - 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 11.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 839.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 637.T + 9.12e5T^{2} \) |
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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
−10.79362011636077056360559561261, −9.808679348015717556555181655322, −8.785405480726387599516454570238, −7.82202061269941747277692550723, −6.46557243205676936061600349675, −6.04793192761779134497314478525, −4.40433686169121043609948429682, −3.61100062040112321213645830065, −1.65353758962625653150982339879, 0,
1.65353758962625653150982339879, 3.61100062040112321213645830065, 4.40433686169121043609948429682, 6.04793192761779134497314478525, 6.46557243205676936061600349675, 7.82202061269941747277692550723, 8.785405480726387599516454570238, 9.808679348015717556555181655322, 10.79362011636077056360559561261