| L(s) = 1 | − 4·7-s − 4·13-s + 6·17-s − 2·19-s − 4·31-s + 10·37-s − 4·43-s + 12·47-s + 9·49-s + 6·53-s − 12·59-s + 10·61-s + 4·67-s + 8·73-s + 10·79-s + 6·83-s + 6·89-s + 16·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 24·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 0.718·31-s + 1.64·37-s − 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.824·53-s − 1.56·59-s + 1.28·61-s + 0.488·67-s + 0.936·73-s + 1.12·79-s + 0.658·83-s + 0.635·89-s + 1.67·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 2.20·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.471770497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.471770497\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.55943312442502, −13.15596805431042, −12.63827907886507, −12.17399438509979, −12.05817825880350, −11.17411418909720, −10.64004336438232, −10.14203658710611, −9.695717552477594, −9.404127426481356, −8.884606867838632, −8.105570098912734, −7.589414022797506, −7.248261119139567, −6.507084679087938, −6.229550067740229, −5.482615662149463, −5.172187673046736, −4.287901829386709, −3.767682099793725, −3.242412649304858, −2.612968420640136, −2.166728233864712, −1.063428167745214, −0.4228075684059587,
0.4228075684059587, 1.063428167745214, 2.166728233864712, 2.612968420640136, 3.242412649304858, 3.767682099793725, 4.287901829386709, 5.172187673046736, 5.482615662149463, 6.229550067740229, 6.507084679087938, 7.248261119139567, 7.589414022797506, 8.105570098912734, 8.884606867838632, 9.404127426481356, 9.695717552477594, 10.14203658710611, 10.64004336438232, 11.17411418909720, 12.05817825880350, 12.17399438509979, 12.63827907886507, 13.15596805431042, 13.55943312442502
