| L(s) = 1 | + 3-s − 2·4-s + 7-s + 9-s + 11-s − 2·12-s + 4·16-s + 3·17-s + 19-s + 21-s + 5·23-s + 27-s − 2·28-s − 2·29-s − 2·31-s + 33-s − 2·36-s − 7·37-s + 5·41-s − 6·43-s − 2·44-s + 4·47-s + 4·48-s − 6·49-s + 3·51-s − 53-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 16-s + 0.727·17-s + 0.229·19-s + 0.218·21-s + 1.04·23-s + 0.192·27-s − 0.377·28-s − 0.371·29-s − 0.359·31-s + 0.174·33-s − 1/3·36-s − 1.15·37-s + 0.780·41-s − 0.914·43-s − 0.301·44-s + 0.583·47-s + 0.577·48-s − 6/7·49-s + 0.420·51-s − 0.137·53-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.921656520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.921656520\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.07263404563425, −12.46108075320747, −12.11915874226360, −11.53085010503766, −10.93474894209518, −10.57397242321670, −9.904102436446778, −9.541130161188034, −9.232826022591759, −8.534394734879220, −8.404459974064436, −7.780140545280564, −7.319154380736653, −6.838079683372196, −6.175570620388284, −5.483694060424407, −5.129621845568800, −4.714337510277470, −4.007953894441329, −3.567064303370717, −3.201785086061856, −2.432810711286085, −1.704673805350137, −1.142289774931878, −0.5060113864412147,
0.5060113864412147, 1.142289774931878, 1.704673805350137, 2.432810711286085, 3.201785086061856, 3.567064303370717, 4.007953894441329, 4.714337510277470, 5.129621845568800, 5.483694060424407, 6.175570620388284, 6.838079683372196, 7.319154380736653, 7.780140545280564, 8.404459974064436, 8.534394734879220, 9.232826022591759, 9.541130161188034, 9.904102436446778, 10.57397242321670, 10.93474894209518, 11.53085010503766, 12.11915874226360, 12.46108075320747, 13.07263404563425
