| L(s) = 1 | + 3-s − 2·4-s − 5-s − 2·9-s + 11-s − 2·12-s − 15-s + 4·16-s + 4·17-s + 2·19-s + 2·20-s + 7·23-s − 4·25-s − 5·27-s − 2·29-s − 3·31-s + 33-s + 4·36-s + 11·37-s + 10·41-s − 4·43-s − 2·44-s + 2·45-s − 4·47-s + 4·48-s + 4·51-s + 2·53-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 2/3·9-s + 0.301·11-s − 0.577·12-s − 0.258·15-s + 16-s + 0.970·17-s + 0.458·19-s + 0.447·20-s + 1.45·23-s − 4/5·25-s − 0.962·27-s − 0.371·29-s − 0.538·31-s + 0.174·33-s + 2/3·36-s + 1.80·37-s + 1.56·41-s − 0.609·43-s − 0.301·44-s + 0.298·45-s − 0.583·47-s + 0.577·48-s + 0.560·51-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−14.11208818399677, −13.67903940982960, −13.14352130922719, −12.75749742425766, −12.22493077519056, −11.55911249217224, −11.25849559180498, −10.67809556767050, −9.857869426889416, −9.493198726346858, −9.203100982902724, −8.585100066074766, −8.083136199171635, −7.702505427767226, −7.265117772868510, −6.395726248224715, −5.712128396159799, −5.398977976062232, −4.738167855059272, −4.006602336232324, −3.708545547562089, −3.003262982818630, −2.590325188677636, −1.470510711656053, −0.8613320590840116, 0,
0.8613320590840116, 1.470510711656053, 2.590325188677636, 3.003262982818630, 3.708545547562089, 4.006602336232324, 4.738167855059272, 5.398977976062232, 5.712128396159799, 6.395726248224715, 7.265117772868510, 7.702505427767226, 8.083136199171635, 8.585100066074766, 9.203100982902724, 9.493198726346858, 9.857869426889416, 10.67809556767050, 11.25849559180498, 11.55911249217224, 12.22493077519056, 12.75749742425766, 13.14352130922719, 13.67903940982960, 14.11208818399677
