| L(s) = 1 | − 5-s − 3·9-s − 11-s + 6·13-s − 6·17-s − 4·19-s − 4·23-s + 25-s + 6·29-s − 10·37-s − 10·41-s − 4·43-s + 3·45-s + 12·47-s + 6·53-s + 55-s − 4·59-s − 14·61-s − 6·65-s + 8·67-s − 16·71-s + 2·73-s + 9·81-s + 12·83-s + 6·85-s + 6·89-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 0.301·11-s + 1.66·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 1.64·37-s − 1.56·41-s − 0.609·43-s + 0.447·45-s + 1.75·47-s + 0.824·53-s + 0.134·55-s − 0.520·59-s − 1.79·61-s − 0.744·65-s + 0.977·67-s − 1.89·71-s + 0.234·73-s + 81-s + 1.31·83-s + 0.650·85-s + 0.635·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6968993365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6968993365\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.90393698659249, −13.90585271172268, −13.64655497235009, −13.45368205474317, −12.48636584634027, −12.08647816775590, −11.59813437994183, −10.94820705467499, −10.63438543039502, −10.24450074801422, −9.155463948014789, −8.700092545385240, −8.497184280229013, −7.976696872905613, −7.107359088555115, −6.480570720916925, −6.183171953114941, −5.464164177669457, −4.789913982528903, −4.105502788326152, −3.614361702284951, −2.904787742496678, −2.192027100460879, −1.436531454924736, −0.2963951810892844,
0.2963951810892844, 1.436531454924736, 2.192027100460879, 2.904787742496678, 3.614361702284951, 4.105502788326152, 4.789913982528903, 5.464164177669457, 6.183171953114941, 6.480570720916925, 7.107359088555115, 7.976696872905613, 8.497184280229013, 8.700092545385240, 9.155463948014789, 10.24450074801422, 10.63438543039502, 10.94820705467499, 11.59813437994183, 12.08647816775590, 12.48636584634027, 13.45368205474317, 13.64655497235009, 13.90585271172268, 14.90393698659249
