| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s − 4·13-s − 2·15-s − 17-s + 2·19-s + 2·21-s − 2·23-s − 25-s − 27-s − 2·29-s + 4·31-s − 4·35-s + 6·37-s + 4·39-s − 6·41-s + 2·43-s + 2·45-s − 3·49-s + 51-s − 12·53-s − 2·57-s + 14·59-s − 6·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s − 0.242·17-s + 0.458·19-s + 0.436·21-s − 0.417·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.676·35-s + 0.986·37-s + 0.640·39-s − 0.937·41-s + 0.304·43-s + 0.298·45-s − 3/7·49-s + 0.140·51-s − 1.64·53-s − 0.264·57-s + 1.82·59-s − 0.768·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.87191787127954, −13.56497235826978, −12.89717119990056, −12.66468591386228, −12.10902076127627, −11.51693384351502, −11.19131377640237, −10.40107028069551, −9.975453500838321, −9.637880727968895, −9.393358670172064, −8.554792066118497, −7.960381009551004, −7.376722750983598, −6.837790358249286, −6.340288041465554, −5.927195506387318, −5.371579972899672, −4.841909976290995, −4.300559511941333, −3.515388598600827, −2.895419645756734, −2.243088342201576, −1.703721717402216, −0.7687036860182527, 0,
0.7687036860182527, 1.703721717402216, 2.243088342201576, 2.895419645756734, 3.515388598600827, 4.300559511941333, 4.841909976290995, 5.371579972899672, 5.927195506387318, 6.340288041465554, 6.837790358249286, 7.376722750983598, 7.960381009551004, 8.554792066118497, 9.393358670172064, 9.637880727968895, 9.975453500838321, 10.40107028069551, 11.19131377640237, 11.51693384351502, 12.10902076127627, 12.66468591386228, 12.89717119990056, 13.56497235826978, 13.87191787127954
