| L(s) = 1 | − 2-s − 2·3-s + 4-s − 5-s + 2·6-s − 5·7-s − 8-s + 9-s + 10-s + 2·11-s − 2·12-s − 13-s + 5·14-s + 2·15-s + 16-s − 17-s − 18-s − 20-s + 10·21-s − 2·22-s − 6·23-s + 2·24-s + 25-s + 26-s + 4·27-s − 5·28-s + 8·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.577·12-s − 0.277·13-s + 1.33·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.223·20-s + 2.18·21-s − 0.426·22-s − 1.25·23-s + 0.408·24-s + 1/5·25-s + 0.196·26-s + 0.769·27-s − 0.944·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−12.86344896648284, −12.42001150937875, −12.18390025355014, −11.80012471655300, −11.37201890666023, −10.68067721901173, −10.31985740903689, −10.00755380313357, −9.552360484706386, −8.902558086031334, −8.633568616122535, −7.984838069397797, −7.254291828693002, −6.943371906951805, −6.359892553604622, −6.245810248135103, −5.670366588212328, −5.065121442565194, −4.326205285739520, −3.840345363816796, −3.297534292533490, −2.640770889496505, −2.157523515226229, −0.9998010612063276, −0.6233374832120870, 0,
0.6233374832120870, 0.9998010612063276, 2.157523515226229, 2.640770889496505, 3.297534292533490, 3.840345363816796, 4.326205285739520, 5.065121442565194, 5.670366588212328, 6.245810248135103, 6.359892553604622, 6.943371906951805, 7.254291828693002, 7.984838069397797, 8.633568616122535, 8.902558086031334, 9.552360484706386, 10.00755380313357, 10.31985740903689, 10.68067721901173, 11.37201890666023, 11.80012471655300, 12.18390025355014, 12.42001150937875, 12.86344896648284
