| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 2·13-s − 2·14-s + 16-s + 3·17-s + 23-s − 5·25-s − 2·26-s + 2·28-s + 3·29-s − 31-s − 32-s − 3·34-s − 7·37-s + 12·41-s − 7·43-s − 46-s − 6·47-s − 3·49-s + 5·50-s + 2·52-s − 6·53-s − 2·56-s − 3·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s + 0.208·23-s − 25-s − 0.392·26-s + 0.377·28-s + 0.557·29-s − 0.179·31-s − 0.176·32-s − 0.514·34-s − 1.15·37-s + 1.87·41-s − 1.06·43-s − 0.147·46-s − 0.875·47-s − 3/7·49-s + 0.707·50-s + 0.277·52-s − 0.824·53-s − 0.267·56-s − 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.55797728378418, −13.13576908661605, −12.48969439324704, −12.06682722276346, −11.57645997038348, −11.12859007078908, −10.77513225121193, −10.20260258423992, −9.648753947131775, −9.384676673203339, −8.620891432474490, −8.164980502745437, −8.006557868301727, −7.311532208659956, −6.801106345440911, −6.293679584067564, −5.623434991748999, −5.274898478614660, −4.581799969108298, −3.939006248061618, −3.381330175285762, −2.761242752722216, −1.992868171972492, −1.504618259215513, −0.9009889269403514, 0,
0.9009889269403514, 1.504618259215513, 1.992868171972492, 2.761242752722216, 3.381330175285762, 3.939006248061618, 4.581799969108298, 5.274898478614660, 5.623434991748999, 6.293679584067564, 6.801106345440911, 7.311532208659956, 8.006557868301727, 8.164980502745437, 8.620891432474490, 9.384676673203339, 9.648753947131775, 10.20260258423992, 10.77513225121193, 11.12859007078908, 11.57645997038348, 12.06682722276346, 12.48969439324704, 13.13576908661605, 13.55797728378418
