| L(s) = 1 | − 2·5-s − 7-s − 4·11-s + 6·17-s − 4·19-s − 25-s − 6·29-s + 2·35-s − 6·37-s − 6·41-s + 4·43-s + 49-s + 2·53-s + 8·55-s + 4·59-s − 2·61-s − 4·67-s + 6·73-s + 4·77-s + 8·79-s + 12·83-s − 12·85-s + 10·89-s + 8·95-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.20·11-s + 1.45·17-s − 0.917·19-s − 1/5·25-s − 1.11·29-s + 0.338·35-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.274·53-s + 1.07·55-s + 0.520·59-s − 0.256·61-s − 0.488·67-s + 0.702·73-s + 0.455·77-s + 0.900·79-s + 1.31·83-s − 1.30·85-s + 1.05·89-s + 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.26705173754494, −13.41287175308051, −13.32846525000668, −12.60599767633097, −12.13514648180982, −11.91717055952358, −11.09826886537407, −10.72345329007258, −10.23725458606558, −9.786353899214011, −9.122922030140424, −8.551828104510117, −8.013440862015510, −7.596818242468455, −7.279606309846872, −6.458086388314862, −5.967900311434953, −5.190674135483298, −5.032187256722447, −3.990612661605488, −3.686762076307848, −3.109273715950716, −2.382660679667937, −1.714118978074714, −0.6634278796342889, 0,
0.6634278796342889, 1.714118978074714, 2.382660679667937, 3.109273715950716, 3.686762076307848, 3.990612661605488, 5.032187256722447, 5.190674135483298, 5.967900311434953, 6.458086388314862, 7.279606309846872, 7.596818242468455, 8.013440862015510, 8.551828104510117, 9.122922030140424, 9.786353899214011, 10.23725458606558, 10.72345329007258, 11.09826886537407, 11.91717055952358, 12.13514648180982, 12.60599767633097, 13.32846525000668, 13.41287175308051, 14.26705173754494
