| L(s) = 1 | − 5-s − 11-s − 2·17-s + 2·19-s − 6·23-s + 25-s + 2·29-s − 6·37-s − 2·41-s + 2·43-s − 2·47-s + 2·53-s + 55-s + 8·59-s − 4·61-s + 4·67-s + 2·71-s + 10·73-s − 4·79-s + 12·83-s + 2·85-s − 2·95-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.485·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.986·37-s − 0.312·41-s + 0.304·43-s − 0.291·47-s + 0.274·53-s + 0.134·55-s + 1.04·59-s − 0.512·61-s + 0.488·67-s + 0.237·71-s + 1.17·73-s − 0.450·79-s + 1.31·83-s + 0.216·85-s − 0.205·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.66645680412384, −12.19509446993940, −11.71978852303698, −11.48738680844461, −10.85178639872309, −10.41544498424272, −10.06941949286551, −9.540565798122147, −9.023336433188699, −8.544843796548500, −8.147909916609121, −7.672182261963267, −7.256442371735263, −6.689580766768581, −6.278265984923305, −5.735229768180975, −5.135247369359823, −4.791975723485997, −4.181600523354901, −3.595409016751718, −3.348126277017752, −2.428955717038743, −2.169699191021874, −1.382447848691140, −0.6607913415840554, 0,
0.6607913415840554, 1.382447848691140, 2.169699191021874, 2.428955717038743, 3.348126277017752, 3.595409016751718, 4.181600523354901, 4.791975723485997, 5.135247369359823, 5.735229768180975, 6.278265984923305, 6.689580766768581, 7.256442371735263, 7.672182261963267, 8.147909916609121, 8.544843796548500, 9.023336433188699, 9.540565798122147, 10.06941949286551, 10.41544498424272, 10.85178639872309, 11.48738680844461, 11.71978852303698, 12.19509446993940, 12.66645680412384
