| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 4·7-s − 8-s + 9-s + 10-s − 12-s + 4·14-s + 15-s + 16-s + 2·17-s − 18-s − 19-s − 20-s + 4·21-s + 8·23-s + 24-s + 25-s − 27-s − 4·28-s + 2·29-s − 30-s + 8·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.872·21-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.755·28-s + 0.371·29-s − 0.182·30-s + 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.01036854007495, −13.33837309459179, −12.99960819428602, −12.43339990358179, −12.10421625642889, −11.50707328055651, −11.13643809310426, −10.39412852185476, −10.06385254782942, −9.791938976795475, −9.058152004978394, −8.542753946717021, −8.224546478145325, −7.200058708422513, −7.079693290076309, −6.488149749980914, −6.176311561865152, −5.233893156574911, −5.004566195318757, −4.052940399303082, −3.371571076221044, −3.086207003640089, −2.312390091373828, −1.334859914550082, −0.6923157483909710, 0,
0.6923157483909710, 1.334859914550082, 2.312390091373828, 3.086207003640089, 3.371571076221044, 4.052940399303082, 5.004566195318757, 5.233893156574911, 6.176311561865152, 6.488149749980914, 7.079693290076309, 7.200058708422513, 8.224546478145325, 8.542753946717021, 9.058152004978394, 9.791938976795475, 10.06385254782942, 10.39412852185476, 11.13643809310426, 11.50707328055651, 12.10421625642889, 12.43339990358179, 12.99960819428602, 13.33837309459179, 14.01036854007495
