| L(s) = 1 | + 3-s − 2·4-s + 5-s + 9-s − 2·12-s + 3·13-s + 15-s + 4·16-s + 3·17-s + 5·19-s − 2·20-s + 4·23-s − 4·25-s + 27-s − 3·29-s − 2·36-s + 4·37-s + 3·39-s + 12·41-s − 7·43-s + 45-s + 7·47-s + 4·48-s − 7·49-s + 3·51-s − 6·52-s − 10·53-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s − 0.577·12-s + 0.832·13-s + 0.258·15-s + 16-s + 0.727·17-s + 1.14·19-s − 0.447·20-s + 0.834·23-s − 4/5·25-s + 0.192·27-s − 0.557·29-s − 1/3·36-s + 0.657·37-s + 0.480·39-s + 1.87·41-s − 1.06·43-s + 0.149·45-s + 1.02·47-s + 0.577·48-s − 49-s + 0.420·51-s − 0.832·52-s − 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 537 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 537 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.620510579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.620510579\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 179 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−10.64030618201326902990685632180, −9.586110307756053535539478295527, −9.286729364424769655691391910301, −8.208911964386550975631737047964, −7.49960104891407745383090791238, −6.06246575378433381864574309523, −5.20005402513594735661113433485, −4.01668844721035844284238592589, −3.05427112716305372657210417661, −1.27781488517096422471696058348,
1.27781488517096422471696058348, 3.05427112716305372657210417661, 4.01668844721035844284238592589, 5.20005402513594735661113433485, 6.06246575378433381864574309523, 7.49960104891407745383090791238, 8.208911964386550975631737047964, 9.286729364424769655691391910301, 9.586110307756053535539478295527, 10.64030618201326902990685632180
