| L(s) = 1 | − 2·7-s + 2·11-s + 13-s + 17-s − 4·19-s − 8·23-s − 2·29-s − 6·31-s − 8·37-s − 4·41-s + 4·43-s − 3·49-s − 14·53-s − 8·59-s − 10·61-s + 12·67-s + 6·71-s + 8·73-s − 4·77-s + 4·79-s + 4·83-s − 2·89-s − 2·91-s + 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.603·11-s + 0.277·13-s + 0.242·17-s − 0.917·19-s − 1.66·23-s − 0.371·29-s − 1.07·31-s − 1.31·37-s − 0.624·41-s + 0.609·43-s − 3/7·49-s − 1.92·53-s − 1.04·59-s − 1.28·61-s + 1.46·67-s + 0.712·71-s + 0.936·73-s − 0.455·77-s + 0.450·79-s + 0.439·83-s − 0.211·89-s − 0.209·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.69103400328528, −12.39856109370180, −12.22686060405539, −11.52414391452178, −10.91485569216705, −10.82336503252817, −10.10490105319176, −9.700185140135926, −9.345423674369016, −8.875015164666864, −8.357834707994823, −7.869140875340848, −7.492106470784442, −6.741508132358422, −6.435446121110512, −6.128760977154420, −5.517059235021240, −5.000226409799746, −4.374461845208960, −3.796337165532000, −3.537285966624237, −2.997838150133502, −2.094364606796266, −1.852681431608361, −1.134301541545260, 0, 0,
1.134301541545260, 1.852681431608361, 2.094364606796266, 2.997838150133502, 3.537285966624237, 3.796337165532000, 4.374461845208960, 5.000226409799746, 5.517059235021240, 6.128760977154420, 6.435446121110512, 6.741508132358422, 7.492106470784442, 7.869140875340848, 8.357834707994823, 8.875015164666864, 9.345423674369016, 9.700185140135926, 10.10490105319176, 10.82336503252817, 10.91485569216705, 11.52414391452178, 12.22686060405539, 12.39856109370180, 12.69103400328528
