| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 2·11-s − 12-s − 15-s + 16-s − 17-s − 18-s + 2·19-s + 20-s − 2·22-s − 2·23-s + 24-s + 25-s − 27-s − 2·29-s + 30-s − 32-s − 2·33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.426·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.348·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.20649803977565, −13.65585757297579, −13.04869908543646, −12.68315704597392, −11.95612095989551, −11.70392131415163, −11.09933680852976, −10.72772130154916, −10.11138054696800, −9.625679360690653, −9.321427788983558, −8.687015265517496, −8.165292022213955, −7.464202749629514, −7.150441545332719, −6.353388639248884, −6.137039778100385, −5.541069278736797, −4.837085728895615, −4.313108438885710, −3.532582592829836, −2.926042608814919, −2.086628548341341, −1.554095020708446, −0.8666273163049122, 0,
0.8666273163049122, 1.554095020708446, 2.086628548341341, 2.926042608814919, 3.532582592829836, 4.313108438885710, 4.837085728895615, 5.541069278736797, 6.137039778100385, 6.353388639248884, 7.150441545332719, 7.464202749629514, 8.165292022213955, 8.687015265517496, 9.321427788983558, 9.625679360690653, 10.11138054696800, 10.72772130154916, 11.09933680852976, 11.70392131415163, 11.95612095989551, 12.68315704597392, 13.04869908543646, 13.65585757297579, 14.20649803977565
