| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·7-s + 8-s + 9-s − 3·11-s + 12-s + 7·13-s + 3·14-s + 16-s + 6·17-s + 18-s + 3·21-s − 3·22-s + 3·23-s + 24-s + 7·26-s + 27-s + 3·28-s + 3·29-s − 2·31-s + 32-s − 3·33-s + 6·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 1.94·13-s + 0.801·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.654·21-s − 0.639·22-s + 0.625·23-s + 0.204·24-s + 1.37·26-s + 0.192·27-s + 0.566·28-s + 0.557·29-s − 0.359·31-s + 0.176·32-s − 0.522·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.828897359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.828897359\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 283 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−14.70313449909785, −14.16996673943942, −13.62045983789272, −13.37908851323667, −12.77691234416465, −12.16225746357793, −11.63973363595410, −11.07315358247591, −10.54702213423592, −10.32046778995143, −9.274780860417632, −8.855276431900721, −8.135782738656704, −7.816188540868598, −7.415397980851913, −6.497151710090068, −5.865440625015613, −5.478585303755913, −4.721672290462131, −4.260651549907905, −3.459486268382893, −3.060451201649624, −2.277877179731399, −1.419261172446508, −0.9844221923833824,
0.9844221923833824, 1.419261172446508, 2.277877179731399, 3.060451201649624, 3.459486268382893, 4.260651549907905, 4.721672290462131, 5.478585303755913, 5.865440625015613, 6.497151710090068, 7.415397980851913, 7.816188540868598, 8.135782738656704, 8.855276431900721, 9.274780860417632, 10.32046778995143, 10.54702213423592, 11.07315358247591, 11.63973363595410, 12.16225746357793, 12.77691234416465, 13.37908851323667, 13.62045983789272, 14.16996673943942, 14.70313449909785
