| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 6·11-s − 6·13-s + 15-s − 4·19-s + 4·21-s + 4·23-s + 25-s + 27-s − 2·29-s − 6·31-s − 6·33-s + 4·35-s − 4·37-s − 6·39-s + 8·41-s + 4·43-s + 45-s + 8·47-s + 9·49-s + 10·53-s − 6·55-s − 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s − 1.66·13-s + 0.258·15-s − 0.917·19-s + 0.872·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.07·31-s − 1.04·33-s + 0.676·35-s − 0.657·37-s − 0.960·39-s + 1.24·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s + 1.37·53-s − 0.809·55-s − 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 67 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.64956564492131, −14.08359766790637, −13.45147455381751, −13.06006096507548, −12.50411214307423, −12.15650750027855, −11.30756488864944, −10.80539356855831, −10.51993389350905, −9.967594765957070, −9.322519651818612, −8.764563071986114, −8.333792485408443, −7.687358945717376, −7.343943155342046, −7.030147879690726, −5.792932412711466, −5.476246738974915, −4.869186264272558, −4.558459604986938, −3.760876879405046, −2.768088984128787, −2.269522080765069, −2.101017535728509, −1.021143174098417, 0,
1.021143174098417, 2.101017535728509, 2.269522080765069, 2.768088984128787, 3.760876879405046, 4.558459604986938, 4.869186264272558, 5.476246738974915, 5.792932412711466, 7.030147879690726, 7.343943155342046, 7.687358945717376, 8.333792485408443, 8.764563071986114, 9.322519651818612, 9.967594765957070, 10.51993389350905, 10.80539356855831, 11.30756488864944, 12.15650750027855, 12.50411214307423, 13.06006096507548, 13.45147455381751, 14.08359766790637, 14.64956564492131
