| L(s) = 1 | + 3·3-s + 5-s + 4·7-s + 6·9-s − 6·13-s + 3·15-s − 4·17-s − 2·19-s + 12·21-s − 3·23-s − 4·25-s + 9·27-s − 29-s + 7·31-s + 4·35-s − 11·37-s − 18·39-s − 4·41-s − 4·43-s + 6·45-s − 8·47-s + 9·49-s − 12·51-s + 2·53-s − 6·57-s + 3·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 1.51·7-s + 2·9-s − 1.66·13-s + 0.774·15-s − 0.970·17-s − 0.458·19-s + 2.61·21-s − 0.625·23-s − 4/5·25-s + 1.73·27-s − 0.185·29-s + 1.25·31-s + 0.676·35-s − 1.80·37-s − 2.88·39-s − 0.624·41-s − 0.609·43-s + 0.894·45-s − 1.16·47-s + 9/7·49-s − 1.68·51-s + 0.274·53-s − 0.794·57-s + 0.390·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 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
−14.61592410039119, −14.07701391474464, −13.82811217568157, −13.39883049603385, −12.71032544852644, −12.14501411056364, −11.68597936239600, −11.02259297904276, −10.30988257131382, −9.890659914533593, −9.510766962143243, −8.814096443601940, −8.229124108656420, −8.176991894575529, −7.468848598012820, −6.924801547115107, −6.406614818867721, −5.158054696205257, −5.070287635336593, −4.266945153874559, −3.834314527405157, −2.933455140787257, −2.223558085301079, −2.050988310444766, −1.422056687058371, 0,
1.422056687058371, 2.050988310444766, 2.223558085301079, 2.933455140787257, 3.834314527405157, 4.266945153874559, 5.070287635336593, 5.158054696205257, 6.406614818867721, 6.924801547115107, 7.468848598012820, 8.176991894575529, 8.229124108656420, 8.814096443601940, 9.510766962143243, 9.890659914533593, 10.30988257131382, 11.02259297904276, 11.68597936239600, 12.14501411056364, 12.71032544852644, 13.39883049603385, 13.82811217568157, 14.07701391474464, 14.61592410039119
