| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s − 2·13-s − 15-s + 6·17-s + 19-s + 25-s + 27-s + 2·29-s + 8·31-s + 4·33-s − 10·37-s − 2·39-s − 2·41-s − 4·43-s − 45-s + 6·51-s − 6·53-s − 4·55-s + 57-s − 10·61-s + 2·65-s + 4·67-s + 8·71-s − 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s + 0.229·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.696·33-s − 1.64·37-s − 0.320·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s + 0.840·51-s − 0.824·53-s − 0.539·55-s + 0.132·57-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + 0.949·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 223440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.506235859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.506235859\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−12.87625046191691, −12.34520331541460, −11.99774632909440, −11.83295708942368, −11.14971510602644, −10.51750817096914, −10.04568291373229, −9.736464086268692, −9.186862180088769, −8.684221778584487, −8.259263079378283, −7.771602645982556, −7.318980284539295, −6.813844182854954, −6.344496251108024, −5.794679925781643, −4.976330366675064, −4.756867785838445, −4.040774422850266, −3.399584527075368, −3.262450177791084, −2.505428757398449, −1.705347202657691, −1.245438824127397, −0.5259304434021211,
0.5259304434021211, 1.245438824127397, 1.705347202657691, 2.505428757398449, 3.262450177791084, 3.399584527075368, 4.040774422850266, 4.756867785838445, 4.976330366675064, 5.794679925781643, 6.344496251108024, 6.813844182854954, 7.318980284539295, 7.771602645982556, 8.259263079378283, 8.684221778584487, 9.186862180088769, 9.736464086268692, 10.04568291373229, 10.51750817096914, 11.14971510602644, 11.83295708942368, 11.99774632909440, 12.34520331541460, 12.87625046191691
