| L(s) = 1 | − 2·3-s + 7-s + 9-s + 2·17-s − 2·19-s − 2·21-s + 8·23-s + 4·27-s − 2·29-s − 4·31-s − 6·37-s − 2·41-s − 8·43-s − 4·47-s + 49-s − 4·51-s − 10·53-s + 4·57-s + 6·59-s − 4·61-s + 63-s + 12·67-s − 16·69-s + 14·73-s + 8·79-s − 11·81-s − 6·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.485·17-s − 0.458·19-s − 0.436·21-s + 1.66·23-s + 0.769·27-s − 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.312·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s + 0.529·57-s + 0.781·59-s − 0.512·61-s + 0.125·63-s + 1.46·67-s − 1.92·69-s + 1.63·73-s + 0.900·79-s − 1.22·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.90475646469343, −16.33727193100600, −15.74534288394970, −14.96312058181721, −14.67077854397963, −13.88909169493861, −13.22408990065117, −12.58760964258088, −12.17667448176048, −11.43874508266689, −11.02744802369731, −10.63868525528126, −9.846690515690665, −9.195727315079806, −8.462614760937241, −7.893869628484495, −6.912684729951710, −6.671779996621094, −5.783272668131190, −5.090806433591382, −4.907627121159729, −3.772405011690071, −3.063283644422747, −1.941198953214486, −1.047577029267884, 0,
1.047577029267884, 1.941198953214486, 3.063283644422747, 3.772405011690071, 4.907627121159729, 5.090806433591382, 5.783272668131190, 6.671779996621094, 6.912684729951710, 7.893869628484495, 8.462614760937241, 9.195727315079806, 9.846690515690665, 10.63868525528126, 11.02744802369731, 11.43874508266689, 12.17667448176048, 12.58760964258088, 13.22408990065117, 13.88909169493861, 14.67077854397963, 14.96312058181721, 15.74534288394970, 16.33727193100600, 16.90475646469343
