| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 15-s + 6·17-s + 4·19-s − 21-s + 25-s + 27-s + 6·29-s − 8·31-s + 35-s − 8·37-s + 6·41-s − 10·43-s − 45-s + 12·47-s + 49-s + 6·51-s + 4·57-s − 6·59-s + 2·61-s − 63-s − 8·67-s + 12·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.169·35-s − 1.31·37-s + 0.937·41-s − 1.52·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.529·57-s − 0.781·59-s + 0.256·61-s − 0.125·63-s − 0.977·67-s + 1.42·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.17506799000585, −14.04186521670179, −13.51658172796236, −12.71114808263191, −12.44378848116029, −12.00591811034205, −11.42382190870590, −10.80926039681191, −10.17672751025704, −9.923516926338917, −9.218091060650307, −8.816491899689763, −8.223965201543077, −7.659078934959772, −7.274336225746931, −6.788455755165077, −5.992603130621987, −5.416069631068858, −4.964184254807181, −4.075452054004290, −3.637976359303458, −3.087792726027770, −2.595114876824100, −1.587461474727728, −1.029938259641019, 0,
1.029938259641019, 1.587461474727728, 2.595114876824100, 3.087792726027770, 3.637976359303458, 4.075452054004290, 4.964184254807181, 5.416069631068858, 5.992603130621987, 6.788455755165077, 7.274336225746931, 7.659078934959772, 8.223965201543077, 8.816491899689763, 9.218091060650307, 9.923516926338917, 10.17672751025704, 10.80926039681191, 11.42382190870590, 12.00591811034205, 12.44378848116029, 12.71114808263191, 13.51658172796236, 14.04186521670179, 14.17506799000585
