| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s − 2·11-s + 16-s − 17-s − 3·19-s + 3·20-s + 2·22-s + 5·23-s + 4·25-s − 6·29-s − 32-s + 34-s + 3·37-s + 3·38-s − 3·40-s − 43-s − 2·44-s − 5·46-s + 6·47-s − 4·50-s + 6·53-s − 6·55-s + 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s − 0.603·11-s + 1/4·16-s − 0.242·17-s − 0.688·19-s + 0.670·20-s + 0.426·22-s + 1.04·23-s + 4/5·25-s − 1.11·29-s − 0.176·32-s + 0.171·34-s + 0.493·37-s + 0.486·38-s − 0.474·40-s − 0.152·43-s − 0.301·44-s − 0.737·46-s + 0.875·47-s − 0.565·50-s + 0.824·53-s − 0.809·55-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.798771552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798771552\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−16.30442670454592, −15.41605956897958, −15.06243412836935, −14.45695512560977, −13.75100783806929, −13.15384796890490, −12.93106164346666, −12.09793777212632, −11.37255721505961, −10.67784051032188, −10.46076081847915, −9.691004864481392, −9.222973552238763, −8.775117318637834, −8.012818170679303, −7.339761299742857, −6.717483178229709, −6.068322272505175, −5.526136703191188, −4.902167745602219, −3.927742905513748, −2.920918300296958, −2.282736226898327, −1.686469652818581, −0.6449703231318714,
0.6449703231318714, 1.686469652818581, 2.282736226898327, 2.920918300296958, 3.927742905513748, 4.902167745602219, 5.526136703191188, 6.068322272505175, 6.717483178229709, 7.339761299742857, 8.012818170679303, 8.775117318637834, 9.222973552238763, 9.691004864481392, 10.46076081847915, 10.67784051032188, 11.37255721505961, 12.09793777212632, 12.93106164346666, 13.15384796890490, 13.75100783806929, 14.45695512560977, 15.06243412836935, 15.41605956897958, 16.30442670454592
