| L(s) = 1 | − 2·7-s + 2·11-s + 2·13-s − 16-s − 2·23-s − 2·29-s + 2·41-s + 49-s + 2·59-s + 2·61-s − 4·77-s − 81-s − 4·83-s − 2·89-s − 4·91-s + 2·97-s − 2·103-s + 2·112-s + 121-s + 127-s + 131-s + 137-s + 139-s + 4·143-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·7-s + 2·11-s + 2·13-s − 16-s − 2·23-s − 2·29-s + 2·41-s + 49-s + 2·59-s + 2·61-s − 4·77-s − 81-s − 4·83-s − 2·89-s − 4·91-s + 2·97-s − 2·103-s + 2·112-s + 121-s + 127-s + 131-s + 137-s + 139-s + 4·143-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5388933796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5388933796\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 283 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2^2$ | \( 1 + T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + T^{4} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.47105597613220777211149229066, −11.58219006169165744610692060487, −11.39136675263032134582558647840, −11.20085922411613741128934633187, −10.28001116438507809233334943749, −9.693425451728701652986693020310, −9.643362890406966006321985614313, −9.018906507862609046506322422309, −8.653637518830049935763110848196, −8.157220923687053708219498646763, −7.05568083941234697100906010314, −7.00139444185545752330422739223, −6.23875193646694958971335889960, −6.03694925877457915720953650028, −5.62659148225833184489551741695, −4.08369552296286370575299516695, −3.98308326875307031073002572314, −3.62059166261996352957104122593, −2.59467010063561747330397905207, −1.51045195254865270478888566135,
1.51045195254865270478888566135, 2.59467010063561747330397905207, 3.62059166261996352957104122593, 3.98308326875307031073002572314, 4.08369552296286370575299516695, 5.62659148225833184489551741695, 6.03694925877457915720953650028, 6.23875193646694958971335889960, 7.00139444185545752330422739223, 7.05568083941234697100906010314, 8.157220923687053708219498646763, 8.653637518830049935763110848196, 9.018906507862609046506322422309, 9.643362890406966006321985614313, 9.693425451728701652986693020310, 10.28001116438507809233334943749, 11.20085922411613741128934633187, 11.39136675263032134582558647840, 11.58219006169165744610692060487, 12.47105597613220777211149229066
