| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 7-s − 8-s + 9-s + 3·10-s + 11-s − 12-s − 4·13-s + 14-s + 3·15-s + 16-s − 3·17-s − 18-s + 19-s − 3·20-s + 21-s − 22-s − 6·23-s + 24-s + 4·25-s + 4·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.229·19-s − 0.670·20-s + 0.218·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s + 4/5·25-s + 0.784·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3873148552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3873148552\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.55812524598011, −14.06550630544924, −13.15777740567444, −12.59464101320762, −12.33124588486989, −11.70410951843409, −11.31711180213061, −11.02071489371444, −10.14355757130571, −9.905602745886183, −9.187442931935134, −8.774071159266377, −7.931034410905328, −7.644382291783724, −7.200253919425717, −6.594015175045699, −5.994844446653938, −5.429196610395210, −4.425526165205929, −4.253954044170133, −3.523669450784638, −2.655207533104903, −2.114067613968557, −0.9916391114494792, −0.3031910306678707,
0.3031910306678707, 0.9916391114494792, 2.114067613968557, 2.655207533104903, 3.523669450784638, 4.253954044170133, 4.425526165205929, 5.429196610395210, 5.994844446653938, 6.594015175045699, 7.200253919425717, 7.644382291783724, 7.931034410905328, 8.774071159266377, 9.187442931935134, 9.905602745886183, 10.14355757130571, 11.02071489371444, 11.31711180213061, 11.70410951843409, 12.33124588486989, 12.59464101320762, 13.15777740567444, 14.06550630544924, 14.55812524598011
