L(s) = 1 | − 5-s + 11-s − 6·13-s − 8·19-s − 6·23-s + 25-s − 4·29-s + 4·31-s + 4·37-s − 2·41-s + 12·43-s − 8·47-s − 4·53-s − 55-s − 10·59-s + 14·61-s + 6·65-s + 4·67-s + 8·71-s − 12·73-s + 2·79-s − 2·83-s + 12·89-s + 8·95-s + 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.301·11-s − 1.66·13-s − 1.83·19-s − 1.25·23-s + 1/5·25-s − 0.742·29-s + 0.718·31-s + 0.657·37-s − 0.312·41-s + 1.82·43-s − 1.16·47-s − 0.549·53-s − 0.134·55-s − 1.30·59-s + 1.79·61-s + 0.744·65-s + 0.488·67-s + 0.949·71-s − 1.40·73-s + 0.225·79-s − 0.219·83-s + 1.27·89-s + 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.77352099278330, −12.15308343316239, −11.89593220169757, −11.40424076422677, −10.75922501779710, −10.58801537798620, −9.881301150009505, −9.538697639039804, −9.218399421916689, −8.390101907220872, −8.173599110914095, −7.748587699369290, −7.119443312771364, −6.826421439234627, −6.134984101731414, −5.903226444798998, −5.084093433057956, −4.685287816215605, −4.166564175617269, −3.890073197804864, −3.109239036522639, −2.422910795052090, −2.194859102204392, −1.478853480994189, −0.5325053342473378, 0,
0.5325053342473378, 1.478853480994189, 2.194859102204392, 2.422910795052090, 3.109239036522639, 3.890073197804864, 4.166564175617269, 4.685287816215605, 5.084093433057956, 5.903226444798998, 6.134984101731414, 6.826421439234627, 7.119443312771364, 7.748587699369290, 8.173599110914095, 8.390101907220872, 9.218399421916689, 9.538697639039804, 9.881301150009505, 10.58801537798620, 10.75922501779710, 11.40424076422677, 11.89593220169757, 12.15308343316239, 12.77352099278330