| L(s) = 1 | + 3·5-s + 4·7-s − 3·11-s − 4·13-s + 7·17-s + 5·19-s − 6·23-s + 4·25-s − 8·29-s − 2·31-s + 12·35-s − 6·37-s − 10·41-s + 5·43-s + 6·47-s + 9·49-s − 6·53-s − 9·55-s − 14·59-s − 11·61-s − 12·65-s + 2·67-s − 5·71-s + 4·73-s − 12·77-s + 4·79-s − 83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.51·7-s − 0.904·11-s − 1.10·13-s + 1.69·17-s + 1.14·19-s − 1.25·23-s + 4/5·25-s − 1.48·29-s − 0.359·31-s + 2.02·35-s − 0.986·37-s − 1.56·41-s + 0.762·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.21·55-s − 1.82·59-s − 1.40·61-s − 1.48·65-s + 0.244·67-s − 0.593·71-s + 0.468·73-s − 1.36·77-s + 0.450·79-s − 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 229 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−15.21600685445215, −14.56172346536008, −14.15051620193921, −13.89754451912730, −13.37277456430916, −12.51970807422039, −12.14195173238886, −11.70541057798485, −10.86167699534396, −10.46222840584292, −9.927275247355723, −9.502782272633463, −8.941603865912631, −8.013164673694892, −7.630292806476434, −7.432025083754555, −6.321222206153207, −5.627104119734480, −5.202632755419401, −5.081853100643870, −4.033844288721706, −3.165602020562072, −2.460462349774385, −1.706890730037031, −1.420998978706287, 0,
1.420998978706287, 1.706890730037031, 2.460462349774385, 3.165602020562072, 4.033844288721706, 5.081853100643870, 5.202632755419401, 5.627104119734480, 6.321222206153207, 7.432025083754555, 7.630292806476434, 8.013164673694892, 8.941603865912631, 9.502782272633463, 9.927275247355723, 10.46222840584292, 10.86167699534396, 11.70541057798485, 12.14195173238886, 12.51970807422039, 13.37277456430916, 13.89754451912730, 14.15051620193921, 14.56172346536008, 15.21600685445215
