| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 4·11-s + 13-s + 15-s − 17-s − 2·19-s + 21-s − 6·23-s + 25-s − 27-s − 2·29-s + 8·31-s − 4·33-s + 35-s + 8·37-s − 39-s + 6·41-s + 6·43-s − 45-s + 8·47-s + 49-s + 51-s + 4·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.458·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.169·35-s + 1.31·37-s − 0.160·39-s + 0.937·41-s + 0.914·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.549·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.58678365835376, −12.13222913885180, −11.89992219036093, −11.42398026309440, −10.94875157959684, −10.59654355197172, −9.964311062924416, −9.588166171945738, −9.189889831837098, −8.559874827046695, −8.240916244403573, −7.588280720880746, −7.201249834018110, −6.596474062681151, −6.250331610413005, −5.870859253519056, −5.396867599153288, −4.422105682760145, −4.291812515263444, −3.923512973017294, −3.257016767897192, −2.505777233974547, −2.091510022407075, −1.130919435367105, −0.8295409572238028, 0,
0.8295409572238028, 1.130919435367105, 2.091510022407075, 2.505777233974547, 3.257016767897192, 3.923512973017294, 4.291812515263444, 4.422105682760145, 5.396867599153288, 5.870859253519056, 6.250331610413005, 6.596474062681151, 7.201249834018110, 7.588280720880746, 8.240916244403573, 8.559874827046695, 9.189889831837098, 9.588166171945738, 9.964311062924416, 10.59654355197172, 10.94875157959684, 11.42398026309440, 11.89992219036093, 12.13222913885180, 12.58678365835376
