| L(s) = 1 | + 5-s + 2·7-s − 4·11-s + 5·13-s + 2·17-s + 2·19-s + 6·23-s + 25-s − 3·29-s − 2·31-s + 2·35-s − 37-s + 7·43-s − 3·47-s − 3·49-s + 9·53-s − 4·55-s + 3·59-s + 6·61-s + 5·65-s − 12·71-s + 6·73-s − 8·77-s + 4·79-s − 83-s + 2·85-s − 13·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 1.20·11-s + 1.38·13-s + 0.485·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s − 0.557·29-s − 0.359·31-s + 0.338·35-s − 0.164·37-s + 1.06·43-s − 0.437·47-s − 3/7·49-s + 1.23·53-s − 0.539·55-s + 0.390·59-s + 0.768·61-s + 0.620·65-s − 1.42·71-s + 0.702·73-s − 0.911·77-s + 0.450·79-s − 0.109·83-s + 0.216·85-s − 1.37·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.623712886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.623712886\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 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
−7.986480265292074962773741402939, −7.38921343011771736374587218336, −6.60249953464733821536703920981, −5.55430506116869532032436401679, −5.42902150316550431561249953514, −4.46721820413265441084032216974, −3.52134866907032351090889184321, −2.74416633538060800282655846389, −1.76068487706322682193547415773, −0.877910559083362156881609995250,
0.877910559083362156881609995250, 1.76068487706322682193547415773, 2.74416633538060800282655846389, 3.52134866907032351090889184321, 4.46721820413265441084032216974, 5.42902150316550431561249953514, 5.55430506116869532032436401679, 6.60249953464733821536703920981, 7.38921343011771736374587218336, 7.986480265292074962773741402939
