| L(s) = 1 | − 3-s − 2·4-s + 7-s + 9-s − 11-s + 2·12-s − 5·13-s + 4·16-s − 6·17-s + 2·19-s − 21-s + 6·23-s − 27-s − 2·28-s − 3·29-s − 4·31-s + 33-s − 2·36-s − 2·37-s + 5·39-s + 43-s + 2·44-s + 12·47-s − 4·48-s − 6·49-s + 6·51-s + 10·52-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 − 1.38·13-s + 16-s − 1.45·17-s + 0.458·19-s − 0.218·21-s + 1.25·23-s − 0.192·27-s − 0.377·28-s − 0.557·29-s − 0.718·31-s + 0.174·33-s − 1/3·36-s − 0.328·37-s + 0.800·39-s + 0.152·43-s + 0.301·44-s + 1.75·47-s − 0.577·48-s − 6/7·49-s + 0.840·51-s + 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−14.65462712674590, −14.13944619826016, −13.56088940902463, −13.10746244583254, −12.68274519491221, −12.20179832428596, −11.58963328050881, −11.07446204803469, −10.55947323830341, −10.06228223754192, −9.431612679056157, −8.999124037776810, −8.608737504838645, −7.762214751234063, −7.277836693061933, −6.934803902748071, −6.008859522181895, −5.386032680682715, −5.063820175967520, −4.471886491520143, −4.061010655048842, −3.143707497067568, −2.442098051055299, −1.655546455443157, −0.7001231825265573, 0,
0.7001231825265573, 1.655546455443157, 2.442098051055299, 3.143707497067568, 4.061010655048842, 4.471886491520143, 5.063820175967520, 5.386032680682715, 6.008859522181895, 6.934803902748071, 7.277836693061933, 7.762214751234063, 8.608737504838645, 8.999124037776810, 9.431612679056157, 10.06228223754192, 10.55947323830341, 11.07446204803469, 11.58963328050881, 12.20179832428596, 12.68274519491221, 13.10746244583254, 13.56088940902463, 14.13944619826016, 14.65462712674590
