| L(s) = 1 | − 2-s + 3-s − 4-s + 2·5-s − 6-s + 3·8-s + 9-s − 2·10-s − 12-s + 2·15-s − 16-s + 6·17-s − 18-s − 4·19-s − 2·20-s − 8·23-s + 3·24-s − 25-s + 27-s + 10·29-s − 2·30-s − 5·32-s − 6·34-s − 36-s − 6·37-s + 4·38-s + 6·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.365·30-s − 0.883·32-s − 1.02·34-s − 1/6·36-s − 0.986·37-s + 0.648·38-s + 0.948·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.30980588284163, −14.14290627755195, −13.68701855940834, −13.05436584861643, −12.63136865830784, −12.12604139916656, −11.46637665618565, −10.61143067627338, −10.23586598644818, −9.865164977540241, −9.550245449804655, −8.888041656841176, −8.321757959992674, −7.993772403796379, −7.569697452935844, −6.646185188479944, −6.217477082972852, −5.587360750232065, −4.889210419619188, −4.395891741565134, −3.671582284234370, −3.076577741774423, −2.170409052356598, −1.703727638500286, −1.005274603781653, 0,
1.005274603781653, 1.703727638500286, 2.170409052356598, 3.076577741774423, 3.671582284234370, 4.395891741565134, 4.889210419619188, 5.587360750232065, 6.217477082972852, 6.646185188479944, 7.569697452935844, 7.993772403796379, 8.321757959992674, 8.888041656841176, 9.550245449804655, 9.865164977540241, 10.23586598644818, 10.61143067627338, 11.46637665618565, 12.12604139916656, 12.63136865830784, 13.05436584861643, 13.68701855940834, 14.14290627755195, 14.30980588284163
