| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 3·11-s − 4·13-s + 15-s + 5·19-s + 2·21-s + 4·23-s + 25-s − 27-s + 9·29-s + 3·33-s + 2·35-s + 6·37-s + 4·39-s + 5·41-s + 2·43-s − 45-s + 10·47-s − 3·49-s − 2·53-s + 3·55-s − 5·57-s + 7·59-s + 9·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.258·15-s + 1.14·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 0.522·33-s + 0.338·35-s + 0.986·37-s + 0.640·39-s + 0.780·41-s + 0.304·43-s − 0.149·45-s + 1.45·47-s − 3/7·49-s − 0.274·53-s + 0.404·55-s − 0.662·57-s + 0.911·59-s + 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.580204728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580204728\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 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
−14.19944923038450, −13.54763399163853, −13.03186415717176, −12.55094313968446, −12.27074485742469, −11.60776320806145, −11.23969801643028, −10.57310899879843, −10.08908701749563, −9.741325492612735, −9.150379619454883, −8.529807495326069, −7.733415873024700, −7.512290895651359, −6.898575592574056, −6.388906736429857, −5.714577138397466, −5.128539623740782, −4.762029761157823, −4.081567334147565, −3.280835961246470, −2.760122471413481, −2.244969242298457, −0.9112361565510394, −0.5710351618575055,
0.5710351618575055, 0.9112361565510394, 2.244969242298457, 2.760122471413481, 3.280835961246470, 4.081567334147565, 4.762029761157823, 5.128539623740782, 5.714577138397466, 6.388906736429857, 6.898575592574056, 7.512290895651359, 7.733415873024700, 8.529807495326069, 9.150379619454883, 9.741325492612735, 10.08908701749563, 10.57310899879843, 11.23969801643028, 11.60776320806145, 12.27074485742469, 12.55094313968446, 13.03186415717176, 13.54763399163853, 14.19944923038450
