| 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.ac |
| 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.b |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 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.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 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.ag |
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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
−14.14490944660098, −13.81642707925681, −13.44458916839924, −12.89514521788092, −12.50183009495440, −11.82278320694662, −11.20608422376669, −10.94217152610384, −10.25853551370692, −9.684186836390033, −9.354451721900464, −8.951617891793972, −8.241025060205446, −7.564597061082342, −7.250461930644828, −6.381121320594208, −6.032748880309532, −5.554406558309671, −5.064048125410311, −4.209070253852885, −3.525756718241672, −3.198806480302114, −2.265583718416501, −1.681713675397061, −1.061566477890500, 0,
1.061566477890500, 1.681713675397061, 2.265583718416501, 3.198806480302114, 3.525756718241672, 4.209070253852885, 5.064048125410311, 5.554406558309671, 6.032748880309532, 6.381121320594208, 7.250461930644828, 7.564597061082342, 8.241025060205446, 8.951617891793972, 9.354451721900464, 9.684186836390033, 10.25853551370692, 10.94217152610384, 11.20608422376669, 11.82278320694662, 12.50183009495440, 12.89514521788092, 13.44458916839924, 13.81642707925681, 14.14490944660098
