| L(s) = 1 | − 3-s − 2·5-s + 9-s − 11-s − 13-s + 2·15-s − 4·17-s + 19-s + 2·23-s − 25-s − 27-s − 6·29-s − 5·31-s + 33-s + 7·37-s + 39-s − 3·43-s − 2·45-s − 8·47-s + 4·51-s − 12·53-s + 2·55-s − 57-s − 10·59-s − 10·61-s + 2·65-s − 3·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.516·15-s − 0.970·17-s + 0.229·19-s + 0.417·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.898·31-s + 0.174·33-s + 1.15·37-s + 0.160·39-s − 0.457·43-s − 0.298·45-s − 1.16·47-s + 0.560·51-s − 1.64·53-s + 0.269·55-s − 0.132·57-s − 1.30·59-s − 1.28·61-s + 0.248·65-s − 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−15.03319973276004, −14.68986376796689, −13.92788612651403, −13.31460016035538, −12.86497203611231, −12.51361719766798, −11.72060157350521, −11.45264550718787, −11.01347538747228, −10.53149787885900, −9.818975388514362, −9.263974370408206, −8.839598809582535, −7.960021389075466, −7.651244525998883, −7.185287678992993, −6.457943692787810, −5.978769887216576, −5.300529208592367, −4.598996798586402, −4.338441600253682, −3.470558452542389, −2.972104655066766, −2.006948350364429, −1.339480236996260, 0, 0,
1.339480236996260, 2.006948350364429, 2.972104655066766, 3.470558452542389, 4.338441600253682, 4.598996798586402, 5.300529208592367, 5.978769887216576, 6.457943692787810, 7.185287678992993, 7.651244525998883, 7.960021389075466, 8.839598809582535, 9.263974370408206, 9.818975388514362, 10.53149787885900, 11.01347538747228, 11.45264550718787, 11.72060157350521, 12.51361719766798, 12.86497203611231, 13.31460016035538, 13.92788612651403, 14.68986376796689, 15.03319973276004
