| L(s) = 1 | − 106.·5-s − 49·7-s − 250.·11-s − 300.·13-s − 2.02e3·17-s + 2.25e3·19-s + 3.09e3·23-s + 8.19e3·25-s + 6.60e3·29-s − 833.·31-s + 5.21e3·35-s + 8.95e3·37-s + 7.20e3·41-s − 1.44e4·43-s − 1.79e4·47-s + 2.40e3·49-s + 1.58e4·53-s + 2.66e4·55-s − 2.67e4·59-s − 2.67e4·61-s + 3.19e4·65-s + 4.44e4·67-s − 2.04e4·71-s + 3.87e4·73-s + 1.22e4·77-s + 6.72e4·79-s − 3.58e4·83-s + ⋯ |
| L(s) = 1 | − 1.90·5-s − 0.377·7-s − 0.623·11-s − 0.493·13-s − 1.69·17-s + 1.43·19-s + 1.21·23-s + 2.62·25-s + 1.45·29-s − 0.155·31-s + 0.719·35-s + 1.07·37-s + 0.669·41-s − 1.19·43-s − 1.18·47-s + 0.142·49-s + 0.773·53-s + 1.18·55-s − 0.998·59-s − 0.922·61-s + 0.938·65-s + 1.21·67-s − 0.480·71-s + 0.850·73-s + 0.235·77-s + 1.21·79-s − 0.571·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 5 | \( 1 + 106.T + 3.12e3T^{2} \) |
| 11 | \( 1 + 250.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 300.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.02e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.25e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 833.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.20e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.44e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.04e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.81e4T + 8.58e9T^{2} \) |
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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
−8.673426650838941802186296450483, −7.933936027181014797261657516363, −7.22114167522106885735100642756, −6.55974272170554742626207304015, −5.00422136167680257886338900566, −4.49046138470818933952966955484, −3.37137892573862412966444770528, −2.68470075078786629601185064226, −0.861559162196326740773649012234, 0,
0.861559162196326740773649012234, 2.68470075078786629601185064226, 3.37137892573862412966444770528, 4.49046138470818933952966955484, 5.00422136167680257886338900566, 6.55974272170554742626207304015, 7.22114167522106885735100642756, 7.933936027181014797261657516363, 8.673426650838941802186296450483
