| L(s) = 1 | + 3·3-s − 7-s + 6·9-s − 6·13-s + 5·17-s − 2·19-s − 3·21-s + 2·23-s + 9·27-s + 6·29-s + 3·31-s − 5·37-s − 18·39-s + 6·41-s − 11·43-s + 47-s + 49-s + 15·51-s + 53-s − 6·57-s + 15·59-s − 10·61-s − 6·63-s + 10·67-s + 6·69-s + 10·71-s − 5·73-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s − 1.66·13-s + 1.21·17-s − 0.458·19-s − 0.654·21-s + 0.417·23-s + 1.73·27-s + 1.11·29-s + 0.538·31-s − 0.821·37-s − 2.88·39-s + 0.937·41-s − 1.67·43-s + 0.145·47-s + 1/7·49-s + 2.10·51-s + 0.137·53-s − 0.794·57-s + 1.95·59-s − 1.28·61-s − 0.755·63-s + 1.22·67-s + 0.722·69-s + 1.18·71-s − 0.585·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.659266862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.659266862\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.01384909943117, −13.56234765179267, −13.01137871825108, −12.51570130513898, −12.16193388757864, −11.65886200611799, −10.68123520513674, −10.21674094539217, −9.803579254913013, −9.480484338634714, −8.924118753948527, −8.184833943838449, −8.105718782719801, −7.420200616449273, −6.866616807165176, −6.573923289427496, −5.475739703725313, −5.046612738356018, −4.352547632779561, −3.793119612568229, −3.138733924635492, −2.736660960897601, −2.238268416818188, −1.498158900754265, −0.6087643375711618,
0.6087643375711618, 1.498158900754265, 2.238268416818188, 2.736660960897601, 3.138733924635492, 3.793119612568229, 4.352547632779561, 5.046612738356018, 5.475739703725313, 6.573923289427496, 6.866616807165176, 7.420200616449273, 8.105718782719801, 8.184833943838449, 8.924118753948527, 9.480484338634714, 9.803579254913013, 10.21674094539217, 10.68123520513674, 11.65886200611799, 12.16193388757864, 12.51570130513898, 13.01137871825108, 13.56234765179267, 14.01384909943117
