| L(s) = 1 | + 2·7-s + 11-s + 4·13-s + 4·17-s − 4·23-s + 6·29-s − 8·31-s + 2·37-s + 10·41-s − 10·43-s − 12·47-s − 3·49-s + 6·53-s + 12·59-s + 10·61-s + 8·67-s − 4·73-s + 2·77-s + 4·79-s + 2·83-s + 10·89-s + 8·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s + 1.10·13-s + 0.970·17-s − 0.834·23-s + 1.11·29-s − 1.43·31-s + 0.328·37-s + 1.56·41-s − 1.52·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s + 1.56·59-s + 1.28·61-s + 0.977·67-s − 0.468·73-s + 0.227·77-s + 0.450·79-s + 0.219·83-s + 1.05·89-s + 0.838·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.913613138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.913613138\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−15.85511432037589, −14.90286866215786, −14.56488949092010, −14.24017078739841, −13.40473237923224, −13.08851603445373, −12.27828022769911, −11.81934912390799, −11.20013252178955, −10.87945745033233, −9.977730345593864, −9.709585183191490, −8.792574207437516, −8.262298593564026, −7.962662627371525, −7.098317250513902, −6.508341549002898, −5.803110466520395, −5.288204428572703, −4.542773496613180, −3.785063840297848, −3.322780473484934, −2.239316680508191, −1.526347086250867, −0.7532384413812066,
0.7532384413812066, 1.526347086250867, 2.239316680508191, 3.322780473484934, 3.785063840297848, 4.542773496613180, 5.288204428572703, 5.803110466520395, 6.508341549002898, 7.098317250513902, 7.962662627371525, 8.262298593564026, 8.792574207437516, 9.709585183191490, 9.977730345593864, 10.87945745033233, 11.20013252178955, 11.81934912390799, 12.27828022769911, 13.08851603445373, 13.40473237923224, 14.24017078739841, 14.56488949092010, 14.90286866215786, 15.85511432037589
