| L(s) = 1 | − 510.·5-s − 753.·7-s + 7.36e3·11-s − 1.12e3·13-s − 3.52e3·17-s − 2.20e4·19-s + 2.23e4·23-s + 1.82e5·25-s + 2.78e4·29-s + 2.61e5·31-s + 3.84e5·35-s + 4.91e5·37-s − 7.60e5·41-s + 4.78e5·43-s − 6.89e5·47-s − 2.55e5·49-s + 7.23e5·53-s − 3.75e6·55-s − 1.15e6·59-s + 2.43e6·61-s + 5.73e5·65-s − 2.37e6·67-s + 2.16e5·71-s − 1.03e6·73-s − 5.54e6·77-s − 4.13e6·79-s + 4.44e5·83-s + ⋯ |
| L(s) = 1 | − 1.82·5-s − 0.830·7-s + 1.66·11-s − 0.141·13-s − 0.174·17-s − 0.736·19-s + 0.383·23-s + 2.33·25-s + 0.211·29-s + 1.57·31-s + 1.51·35-s + 1.59·37-s − 1.72·41-s + 0.918·43-s − 0.968·47-s − 0.310·49-s + 0.667·53-s − 3.04·55-s − 0.734·59-s + 1.37·61-s + 0.259·65-s − 0.965·67-s + 0.0716·71-s − 0.312·73-s − 1.38·77-s − 0.942·79-s + 0.0854·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 510.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 753.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.36e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.12e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.52e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.20e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.23e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.78e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.61e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.91e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.78e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.89e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.23e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.15e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.43e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.37e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.16e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.03e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.44e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.13e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.22e5T + 8.07e13T^{2} \) |
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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
−9.895234930671784007833682051051, −8.863580477206860021399205314070, −8.103281866581148937641048289084, −6.95950414535850296485123612018, −6.38137773061219643222161699254, −4.55364935876674709420569782218, −3.89767460348828513975779369293, −2.94712840989272236799438030878, −1.04871154262923754304690217883, 0,
1.04871154262923754304690217883, 2.94712840989272236799438030878, 3.89767460348828513975779369293, 4.55364935876674709420569782218, 6.38137773061219643222161699254, 6.95950414535850296485123612018, 8.103281866581148937641048289084, 8.863580477206860021399205314070, 9.895234930671784007833682051051
