| L(s) = 1 | + 3-s − 2·4-s − 7-s − 2·9-s − 3·11-s − 2·12-s + 4·13-s + 4·16-s + 6·17-s − 2·19-s − 21-s − 6·23-s − 5·25-s − 5·27-s + 2·28-s + 6·29-s + 4·31-s − 3·33-s + 4·36-s + 37-s + 4·39-s + 9·41-s − 8·43-s + 6·44-s + 4·48-s − 6·49-s + 6·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.577·12-s + 1.10·13-s + 16-s + 1.45·17-s − 0.458·19-s − 0.218·21-s − 1.25·23-s − 25-s − 0.962·27-s + 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.522·33-s + 2/3·36-s + 0.164·37-s + 0.640·39-s + 1.40·41-s − 1.21·43-s + 0.904·44-s + 0.577·48-s − 6/7·49-s + 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 - T \) | |
| 47 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.23651260162356, −13.71166188251753, −13.18627348306374, −13.08114592558208, −12.25062832353473, −11.83260105127406, −11.32134810674225, −10.46415776776442, −10.11100324293829, −9.787273805771055, −9.157713461173347, −8.514066172295321, −8.146765789128445, −7.999474304948237, −7.262434365028973, −6.229867935769922, −5.963301122913841, −5.477535921813826, −4.803667226684541, −4.079279124581626, −3.640255401422552, −3.085804406140456, −2.532191306638304, −1.634736482996229, −0.7836564795851593, 0,
0.7836564795851593, 1.634736482996229, 2.532191306638304, 3.085804406140456, 3.640255401422552, 4.079279124581626, 4.803667226684541, 5.477535921813826, 5.963301122913841, 6.229867935769922, 7.262434365028973, 7.999474304948237, 8.146765789128445, 8.514066172295321, 9.157713461173347, 9.787273805771055, 10.11100324293829, 10.46415776776442, 11.32134810674225, 11.83260105127406, 12.25062832353473, 13.08114592558208, 13.18627348306374, 13.71166188251753, 14.23651260162356
