| L(s) = 1 | − 5-s + 4·7-s − 13-s − 2·17-s − 4·19-s + 25-s + 10·29-s − 2·31-s − 4·35-s − 8·37-s − 12·41-s + 2·43-s − 8·47-s + 9·49-s + 12·53-s + 12·61-s + 65-s + 4·67-s − 16·71-s + 2·73-s − 8·79-s − 6·83-s + 2·85-s − 18·89-s − 4·91-s + 4·95-s − 4·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.277·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 1.85·29-s − 0.359·31-s − 0.676·35-s − 1.31·37-s − 1.87·41-s + 0.304·43-s − 1.16·47-s + 9/7·49-s + 1.64·53-s + 1.53·61-s + 0.124·65-s + 0.488·67-s − 1.89·71-s + 0.234·73-s − 0.900·79-s − 0.658·83-s + 0.216·85-s − 1.90·89-s − 0.419·91-s + 0.410·95-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−12.85640629392544, −12.50345827731600, −11.88506870840960, −11.55622334278880, −11.34273431864400, −10.61718317516559, −10.24146367101005, −10.02929820618172, −9.018871219081047, −8.676108796616697, −8.276178156922337, −8.123348071045194, −7.240270530044673, −6.971618751837509, −6.548431368599753, −5.740639249723809, −5.263916611408409, −4.791960236146413, −4.398831889711038, −3.949494819963881, −3.214384678417844, −2.632795418600703, −1.929545080732409, −1.593785953280984, −0.7797415559835138, 0,
0.7797415559835138, 1.593785953280984, 1.929545080732409, 2.632795418600703, 3.214384678417844, 3.949494819963881, 4.398831889711038, 4.791960236146413, 5.263916611408409, 5.740639249723809, 6.548431368599753, 6.971618751837509, 7.240270530044673, 8.123348071045194, 8.276178156922337, 8.676108796616697, 9.018871219081047, 10.02929820618172, 10.24146367101005, 10.61718317516559, 11.34273431864400, 11.55622334278880, 11.88506870840960, 12.50345827731600, 12.85640629392544
