| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s − 15-s − 3·17-s − 19-s + 3·23-s − 4·25-s − 27-s − 6·31-s + 4·33-s − 11·43-s + 45-s + 3·51-s − 4·53-s − 4·55-s + 57-s − 6·59-s + 10·61-s + 2·67-s − 3·69-s − 4·71-s − 2·73-s + 4·75-s + 6·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.258·15-s − 0.727·17-s − 0.229·19-s + 0.625·23-s − 4/5·25-s − 0.192·27-s − 1.07·31-s + 0.696·33-s − 1.67·43-s + 0.149·45-s + 0.420·51-s − 0.549·53-s − 0.539·55-s + 0.132·57-s − 0.781·59-s + 1.28·61-s + 0.244·67-s − 0.361·69-s − 0.474·71-s − 0.234·73-s + 0.461·75-s + 0.675·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.56628184508781, −13.06393699204413, −12.82960371337922, −12.29611145393084, −11.67773053330776, −11.16969149140905, −10.91374645107986, −10.34068851509342, −9.925901149532192, −9.475880053619030, −8.915130441422813, −8.323698922559483, −7.921873827569910, −7.289059533312598, −6.832644443672682, −6.360311918680450, −5.768618014465085, −5.195079604905753, −5.079221916311332, −4.234201035400015, −3.777313190113177, −2.948277408006130, −2.482296130496233, −1.796419256787900, −1.263283306126657, 0, 0,
1.263283306126657, 1.796419256787900, 2.482296130496233, 2.948277408006130, 3.777313190113177, 4.234201035400015, 5.079221916311332, 5.195079604905753, 5.768618014465085, 6.360311918680450, 6.832644443672682, 7.289059533312598, 7.921873827569910, 8.323698922559483, 8.915130441422813, 9.475880053619030, 9.925901149532192, 10.34068851509342, 10.91374645107986, 11.16969149140905, 11.67773053330776, 12.29611145393084, 12.82960371337922, 13.06393699204413, 13.56628184508781
