| L(s) = 1 | − 2·3-s + 7-s + 9-s + 2·13-s + 5·17-s + 8·19-s − 2·21-s − 23-s + 4·27-s + 5·29-s + 5·31-s + 7·37-s − 4·39-s − 7·41-s − 4·43-s − 2·47-s − 6·49-s − 10·51-s − 53-s − 16·57-s + 3·59-s + 6·61-s + 63-s − 13·67-s + 2·69-s − 13·71-s − 8·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.21·17-s + 1.83·19-s − 0.436·21-s − 0.208·23-s + 0.769·27-s + 0.928·29-s + 0.898·31-s + 1.15·37-s − 0.640·39-s − 1.09·41-s − 0.609·43-s − 0.291·47-s − 6/7·49-s − 1.40·51-s − 0.137·53-s − 2.11·57-s + 0.390·59-s + 0.768·61-s + 0.125·63-s − 1.58·67-s + 0.240·69-s − 1.54·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.10159848032785, −14.67662504937270, −14.00703680368433, −13.63396708203329, −13.06081338155904, −12.21578995422270, −11.89677578939913, −11.63080557244263, −11.04578922441195, −10.40246867374106, −9.965277253988734, −9.489882996080006, −8.636751857816206, −8.054613395697605, −7.667437797950163, −6.806151081853435, −6.407923009226952, −5.660937397387969, −5.345232537542374, −4.784431377918361, −4.071131312553030, −3.175695758464538, −2.764648929964770, −1.262616842172367, −1.209966854866570, 0,
1.209966854866570, 1.262616842172367, 2.764648929964770, 3.175695758464538, 4.071131312553030, 4.784431377918361, 5.345232537542374, 5.660937397387969, 6.407923009226952, 6.806151081853435, 7.667437797950163, 8.054613395697605, 8.636751857816206, 9.489882996080006, 9.965277253988734, 10.40246867374106, 11.04578922441195, 11.63080557244263, 11.89677578939913, 12.21578995422270, 13.06081338155904, 13.63396708203329, 14.00703680368433, 14.67662504937270, 15.10159848032785
