L(s) = 1 | − 2-s + 4-s − 8-s − 11-s + 4·13-s + 16-s − 3·17-s − 2·19-s + 22-s − 23-s − 4·26-s + 29-s − 6·31-s − 32-s + 3·34-s − 4·37-s + 2·38-s + 12·41-s − 4·43-s − 44-s + 46-s + 2·47-s − 7·49-s + 4·52-s − 58-s + 6·59-s − 5·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.301·11-s + 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.458·19-s + 0.213·22-s − 0.208·23-s − 0.784·26-s + 0.185·29-s − 1.07·31-s − 0.176·32-s + 0.514·34-s − 0.657·37-s + 0.324·38-s + 1.87·41-s − 0.609·43-s − 0.150·44-s + 0.147·46-s + 0.291·47-s − 49-s + 0.554·52-s − 0.131·58-s + 0.781·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.54134130439706, −14.29243661439614, −13.49863265987230, −13.04467502497514, −12.68674089543526, −11.96574117784872, −11.41575446073871, −10.88551739412643, −10.68524576118618, −9.993822814491259, −9.357255437926758, −8.979872375350327, −8.342412787974907, −8.081709390100691, −7.241195073358133, −6.897599993361939, −6.114595188967579, −5.845892313817482, −5.027373159544968, −4.320135875659987, −3.704127524257786, −3.066865489777983, −2.251941927497941, −1.724276624935192, −0.8686597040388668, 0,
0.8686597040388668, 1.724276624935192, 2.251941927497941, 3.066865489777983, 3.704127524257786, 4.320135875659987, 5.027373159544968, 5.845892313817482, 6.114595188967579, 6.897599993361939, 7.241195073358133, 8.081709390100691, 8.342412787974907, 8.979872375350327, 9.357255437926758, 9.993822814491259, 10.68524576118618, 10.88551739412643, 11.41575446073871, 11.96574117784872, 12.68674089543526, 13.04467502497514, 13.49863265987230, 14.29243661439614, 14.54134130439706