L(s) = 1 | + 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 1.41i·5-s + 2.00·6-s − 7-s − 1.00·9-s + 2.00·10-s + 11-s + 1.41i·12-s + 13-s − 1.41i·14-s − 2.00·15-s − 0.999·16-s − 1.41i·18-s + 1.41i·19-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 1.41i·5-s + 2.00·6-s − 7-s − 1.00·9-s + 2.00·10-s + 11-s + 1.41i·12-s + 13-s − 1.41i·14-s − 2.00·15-s − 0.999·16-s − 1.41i·18-s + 1.41i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7340935769\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7340935769\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 - T \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 3 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 + 1.41iT - T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 2T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−12.47105597613220777211149229066, −11.58219006169165744610692060487, −9.693425451728701652986693020310, −8.653637518830049935763110848196, −8.157220923687053708219498646763, −7.05568083941234697100906010314, −6.23875193646694958971335889960, −5.62659148225833184489551741695, −3.98308326875307031073002572314, −1.51045195254865270478888566135,
2.59467010063561747330397905207, 3.62059166261996352957104122593, 4.08369552296286370575299516695, 6.03694925877457915720953650028, 7.00139444185545752330422739223, 9.018906507862609046506322422309, 9.643362890406966006321985614313, 10.28001116438507809233334943749, 11.20085922411613741128934633187, 11.39136675263032134582558647840