| L(s) = 1 | − 2·5-s − 7-s + 2·11-s + 13-s + 8·17-s + 4·19-s − 4·23-s − 25-s − 8·29-s + 2·35-s + 2·37-s + 10·41-s + 4·43-s − 2·47-s + 49-s + 4·53-s − 4·55-s + 14·59-s + 2·61-s − 2·65-s − 12·67-s − 10·71-s + 14·73-s − 2·77-s + 10·83-s − 16·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.603·11-s + 0.277·13-s + 1.94·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.48·29-s + 0.338·35-s + 0.328·37-s + 1.56·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 0.549·53-s − 0.539·55-s + 1.82·59-s + 0.256·61-s − 0.248·65-s − 1.46·67-s − 1.18·71-s + 1.63·73-s − 0.227·77-s + 1.09·83-s − 1.73·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.915258746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.915258746\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.29953079539212, −14.74189594515634, −14.31518285198176, −13.81420592096587, −13.08389949171233, −12.55225633892790, −11.98850224808399, −11.65049437468319, −11.14244073143339, −10.37589726368070, −9.751374342470774, −9.437496124061912, −8.678859890098115, −7.993761595144743, −7.460888224785619, −7.267091117890169, −6.131397805739498, −5.818951710552306, −5.147243013390991, −4.157170841271007, −3.724365486830556, −3.296635116513549, −2.347695191469599, −1.317002324201181, −0.5988424148094516,
0.5988424148094516, 1.317002324201181, 2.347695191469599, 3.296635116513549, 3.724365486830556, 4.157170841271007, 5.147243013390991, 5.818951710552306, 6.131397805739498, 7.267091117890169, 7.460888224785619, 7.993761595144743, 8.678859890098115, 9.437496124061912, 9.751374342470774, 10.37589726368070, 11.14244073143339, 11.65049437468319, 11.98850224808399, 12.55225633892790, 13.08389949171233, 13.81420592096587, 14.31518285198176, 14.74189594515634, 15.29953079539212
