| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 4·11-s − 5·13-s − 15-s + 3·17-s + 19-s − 2·21-s − 23-s − 4·25-s + 27-s − 6·29-s + 9·31-s − 4·33-s + 2·35-s + 4·37-s − 5·39-s − 41-s + 8·43-s − 45-s − 3·49-s + 3·51-s − 6·53-s + 4·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.38·13-s − 0.258·15-s + 0.727·17-s + 0.229·19-s − 0.436·21-s − 0.208·23-s − 4/5·25-s + 0.192·27-s − 1.11·29-s + 1.61·31-s − 0.696·33-s + 0.338·35-s + 0.657·37-s − 0.800·39-s − 0.156·41-s + 1.21·43-s − 0.149·45-s − 3/7·49-s + 0.420·51-s − 0.824·53-s + 0.539·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 23 | \( 1 + T \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.25211881097133, −12.95823943426619, −12.51971570593947, −12.07323555175998, −11.60331204439091, −11.02197895776523, −10.41646743350058, −9.959280238237057, −9.577704932911252, −9.389779199043912, −8.434296666266391, −8.062728728358017, −7.637433292773641, −7.339362449427820, −6.728971983815810, −6.038076967182559, −5.588075475071375, −4.957293313020039, −4.500303575659474, −3.838612350214876, −3.322183352497151, −2.647465573750862, −2.494698282041847, −1.616742312510877, −0.6462549199252677, 0,
0.6462549199252677, 1.616742312510877, 2.494698282041847, 2.647465573750862, 3.322183352497151, 3.838612350214876, 4.500303575659474, 4.957293313020039, 5.588075475071375, 6.038076967182559, 6.728971983815810, 7.339362449427820, 7.637433292773641, 8.062728728358017, 8.434296666266391, 9.389779199043912, 9.577704932911252, 9.959280238237057, 10.41646743350058, 11.02197895776523, 11.60331204439091, 12.07323555175998, 12.51971570593947, 12.95823943426619, 13.25211881097133
