| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s + 3·11-s − 4·13-s − 4·14-s + 16-s − 3·17-s + 7·19-s + 3·22-s − 6·23-s − 5·25-s − 4·26-s − 4·28-s + 4·31-s + 32-s − 3·34-s − 8·37-s + 7·38-s − 3·41-s + 7·43-s + 3·44-s − 6·46-s + 6·47-s + 9·49-s − 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 0.904·11-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s + 1.60·19-s + 0.639·22-s − 1.25·23-s − 25-s − 0.784·26-s − 0.755·28-s + 0.718·31-s + 0.176·32-s − 0.514·34-s − 1.31·37-s + 1.13·38-s − 0.468·41-s + 1.06·43-s + 0.452·44-s − 0.884·46-s + 0.875·47-s + 9/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.74110316557556, −13.36758350824724, −12.52902100025109, −12.17859765003931, −12.10018003616177, −11.48866271575011, −10.80539947751641, −10.23180511395559, −9.820158210713382, −9.280231572310277, −9.193543871566874, −8.135541315930336, −7.717129470011200, −7.011416678261196, −6.807458062844924, −6.201206264536175, −5.724340654003779, −5.257192135528618, −4.458183691534498, −4.044848938883256, −3.447340523760080, −3.029775147073470, −2.337544216503624, −1.783438765277703, −0.7880445909368227, 0,
0.7880445909368227, 1.783438765277703, 2.337544216503624, 3.029775147073470, 3.447340523760080, 4.044848938883256, 4.458183691534498, 5.257192135528618, 5.724340654003779, 6.201206264536175, 6.807458062844924, 7.011416678261196, 7.717129470011200, 8.135541315930336, 9.193543871566874, 9.280231572310277, 9.820158210713382, 10.23180511395559, 10.80539947751641, 11.48866271575011, 12.10018003616177, 12.17859765003931, 12.52902100025109, 13.36758350824724, 13.74110316557556
