| L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 3·9-s + 3·11-s − 3·14-s + 16-s + 17-s − 3·18-s + 3·22-s − 3·23-s − 3·28-s + 6·29-s − 4·31-s + 32-s + 34-s − 3·36-s − 4·37-s + 2·41-s + 43-s + 3·44-s − 3·46-s + 6·47-s + 2·49-s + 2·53-s − 3·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 9-s + 0.904·11-s − 0.801·14-s + 1/4·16-s + 0.242·17-s − 0.707·18-s + 0.639·22-s − 0.625·23-s − 0.566·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 1/2·36-s − 0.657·37-s + 0.312·41-s + 0.152·43-s + 0.452·44-s − 0.442·46-s + 0.875·47-s + 2/7·49-s + 0.274·53-s − 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.92870155341362, −12.99205037093591, −12.76813455465214, −12.11068511487736, −11.94152586025826, −11.33780737273329, −10.84285428229080, −10.30169518343374, −9.778303484919876, −9.347604748590208, −8.687844286880968, −8.453340887024839, −7.585431781191399, −7.163519209360568, −6.547714675184726, −6.149432124743678, −5.800375330693658, −5.203699418159641, −4.528069676106058, −3.916738344603309, −3.469260177348592, −2.963139091039205, −2.436374838662288, −1.676904305574195, −0.8283194793552979, 0,
0.8283194793552979, 1.676904305574195, 2.436374838662288, 2.963139091039205, 3.469260177348592, 3.916738344603309, 4.528069676106058, 5.203699418159641, 5.800375330693658, 6.149432124743678, 6.547714675184726, 7.163519209360568, 7.585431781191399, 8.453340887024839, 8.687844286880968, 9.347604748590208, 9.778303484919876, 10.30169518343374, 10.84285428229080, 11.33780737273329, 11.94152586025826, 12.11068511487736, 12.76813455465214, 12.99205037093591, 13.92870155341362
