| L(s) = 1 | − 2·5-s + 7-s + 4·13-s + 6·19-s + 23-s − 25-s + 7·29-s + 6·31-s − 2·35-s + 37-s − 4·41-s + 43-s − 4·47-s + 49-s − 3·53-s − 2·59-s + 10·61-s − 8·65-s − 4·67-s − 71-s − 8·73-s − 9·79-s − 16·89-s + 4·91-s − 12·95-s − 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 1.10·13-s + 1.37·19-s + 0.208·23-s − 1/5·25-s + 1.29·29-s + 1.07·31-s − 0.338·35-s + 0.164·37-s − 0.624·41-s + 0.152·43-s − 0.583·47-s + 1/7·49-s − 0.412·53-s − 0.260·59-s + 1.28·61-s − 0.992·65-s − 0.488·67-s − 0.118·71-s − 0.936·73-s − 1.01·79-s − 1.69·89-s + 0.419·91-s − 1.23·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.13261652790415, −14.02590657044239, −13.34261947453373, −12.91414811664803, −12.17619606784824, −11.74603688098139, −11.46310330254535, −11.01356432291312, −10.20412092500832, −9.999391068689486, −9.183392498931113, −8.605299364094468, −8.241867952304061, −7.751120009580467, −7.231434401646258, −6.595387268238310, −6.085568172979253, −5.390886260567262, −4.834342695838506, −4.249807249482191, −3.693710965612347, −3.106034364616714, −2.544343902471363, −1.387326048502981, −1.060764691867123, 0,
1.060764691867123, 1.387326048502981, 2.544343902471363, 3.106034364616714, 3.693710965612347, 4.249807249482191, 4.834342695838506, 5.390886260567262, 6.085568172979253, 6.595387268238310, 7.231434401646258, 7.751120009580467, 8.241867952304061, 8.605299364094468, 9.183392498931113, 9.999391068689486, 10.20412092500832, 11.01356432291312, 11.46310330254535, 11.74603688098139, 12.17619606784824, 12.91414811664803, 13.34261947453373, 14.02590657044239, 14.13261652790415
