| L(s) = 1 | − 5-s + 2·7-s − 4·11-s − 13-s − 8·17-s − 2·19-s − 4·23-s + 25-s + 8·29-s + 10·31-s − 2·35-s + 6·37-s − 6·41-s − 8·43-s + 8·47-s − 3·49-s + 12·53-s + 4·55-s + 4·59-s + 10·61-s + 65-s + 2·67-s + 6·73-s − 8·77-s + 12·79-s + 4·83-s + 8·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 1.20·11-s − 0.277·13-s − 1.94·17-s − 0.458·19-s − 0.834·23-s + 1/5·25-s + 1.48·29-s + 1.79·31-s − 0.338·35-s + 0.986·37-s − 0.937·41-s − 1.21·43-s + 1.16·47-s − 3/7·49-s + 1.64·53-s + 0.539·55-s + 0.520·59-s + 1.28·61-s + 0.124·65-s + 0.244·67-s + 0.702·73-s − 0.911·77-s + 1.35·79-s + 0.439·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.370708215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370708215\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.392475829969510381401781311959, −7.74238112049599024533718048576, −6.81531531819536411511662295224, −6.27920097741182817948407464345, −5.09604613832881741539687762080, −4.68099993500533399961817521131, −3.94440097116773684012776351687, −2.64389855668014195095493639358, −2.15847934948711881880709546475, −0.62406490911934408422076009247,
0.62406490911934408422076009247, 2.15847934948711881880709546475, 2.64389855668014195095493639358, 3.94440097116773684012776351687, 4.68099993500533399961817521131, 5.09604613832881741539687762080, 6.27920097741182817948407464345, 6.81531531819536411511662295224, 7.74238112049599024533718048576, 8.392475829969510381401781311959
