| L(s) = 1 | − 2·3-s + 9-s − 3·11-s − 13-s + 6·17-s + 19-s − 9·23-s + 4·27-s − 6·29-s + 8·31-s + 6·33-s − 7·37-s + 2·39-s + 3·41-s + 2·43-s − 9·47-s − 12·51-s + 9·53-s − 2·57-s − 8·61-s + 8·67-s + 18·69-s + 4·73-s − 10·79-s − 11·81-s + 12·87-s + 6·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 1.45·17-s + 0.229·19-s − 1.87·23-s + 0.769·27-s − 1.11·29-s + 1.43·31-s + 1.04·33-s − 1.15·37-s + 0.320·39-s + 0.468·41-s + 0.304·43-s − 1.31·47-s − 1.68·51-s + 1.23·53-s − 0.264·57-s − 1.02·61-s + 0.977·67-s + 2.16·69-s + 0.468·73-s − 1.12·79-s − 1.22·81-s + 1.28·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.10983349421917, −13.91156990825035, −13.13919155975550, −12.64747793212383, −12.13467556968657, −11.83026004005437, −11.41526944781041, −10.71456341279410, −10.22577932957523, −10.03093482285472, −9.408050547897066, −8.605631495534373, −7.959311417532249, −7.766548901381123, −7.023288742854130, −6.422680324913551, −5.804080775194989, −5.542558862525742, −5.025088127739004, −4.403181366153745, −3.678395423508429, −3.050699342056741, −2.323819226228537, −1.560551702149027, −0.6928852328906927, 0,
0.6928852328906927, 1.560551702149027, 2.323819226228537, 3.050699342056741, 3.678395423508429, 4.403181366153745, 5.025088127739004, 5.542558862525742, 5.804080775194989, 6.422680324913551, 7.023288742854130, 7.766548901381123, 7.959311417532249, 8.605631495534373, 9.408050547897066, 10.03093482285472, 10.22577932957523, 10.71456341279410, 11.41526944781041, 11.83026004005437, 12.13467556968657, 12.64747793212383, 13.13919155975550, 13.91156990825035, 14.10983349421917
