| L(s) = 1 | + 3-s + 4·5-s + 7-s + 9-s − 6·13-s + 4·15-s − 2·19-s + 21-s − 4·23-s + 11·25-s + 27-s + 2·29-s + 2·31-s + 4·35-s + 2·37-s − 6·39-s − 4·43-s + 4·45-s − 6·47-s + 49-s + 2·53-s − 2·57-s − 14·61-s + 63-s − 24·65-s + 12·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s + 1.03·15-s − 0.458·19-s + 0.218·21-s − 0.834·23-s + 11/5·25-s + 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.676·35-s + 0.328·37-s − 0.960·39-s − 0.609·43-s + 0.596·45-s − 0.875·47-s + 1/7·49-s + 0.274·53-s − 0.264·57-s − 1.79·61-s + 0.125·63-s − 2.97·65-s + 1.46·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−14.13845169643418, −13.82236163888602, −13.40022553121411, −12.70700940090034, −12.46994165910557, −11.85398227340977, −11.16254965060612, −10.50892228516117, −10.02683970167088, −9.803168602240246, −9.319324328778975, −8.754233175035384, −8.203431503853654, −7.642071793293950, −7.048699841964160, −6.446744795852636, −6.059898542805904, −5.306393892850702, −4.897572494664576, −4.422350955655852, −3.523056453942154, −2.659864090602578, −2.408335383693173, −1.812116812522052, −1.203993124006594, 0,
1.203993124006594, 1.812116812522052, 2.408335383693173, 2.659864090602578, 3.523056453942154, 4.422350955655852, 4.897572494664576, 5.306393892850702, 6.059898542805904, 6.446744795852636, 7.048699841964160, 7.642071793293950, 8.203431503853654, 8.754233175035384, 9.319324328778975, 9.803168602240246, 10.02683970167088, 10.50892228516117, 11.16254965060612, 11.85398227340977, 12.46994165910557, 12.70700940090034, 13.40022553121411, 13.82236163888602, 14.13845169643418
