| L(s) = 1 | − 3-s + 9-s − 4·11-s + 4·13-s + 2·17-s + 19-s − 6·23-s − 5·25-s − 27-s + 2·29-s − 6·31-s + 4·33-s − 12·37-s − 4·39-s − 2·41-s − 4·43-s + 10·47-s − 2·51-s − 6·53-s − 57-s + 12·59-s − 2·61-s + 8·67-s + 6·69-s − 8·71-s + 2·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 0.485·17-s + 0.229·19-s − 1.25·23-s − 25-s − 0.192·27-s + 0.371·29-s − 1.07·31-s + 0.696·33-s − 1.97·37-s − 0.640·39-s − 0.312·41-s − 0.609·43-s + 1.45·47-s − 0.280·51-s − 0.824·53-s − 0.132·57-s + 1.56·59-s − 0.256·61-s + 0.977·67-s + 0.722·69-s − 0.949·71-s + 0.234·73-s + 0.577·75-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 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.45747607584295, −12.83046251347308, −12.48117225425723, −11.98508172673626, −11.51353455087903, −11.06525475496086, −10.46458797534281, −10.27316579916380, −9.765957505993618, −9.137849938013737, −8.491483644249331, −8.182517542175901, −7.605451171392957, −7.172253721540431, −6.546125095704613, −6.011820435049564, −5.413154400838714, −5.370943413063020, −4.544744014199003, −3.769055289918347, −3.615818453820018, −2.790316234516145, −1.995472858521880, −1.613688223429531, −0.6727712430544866, 0,
0.6727712430544866, 1.613688223429531, 1.995472858521880, 2.790316234516145, 3.615818453820018, 3.769055289918347, 4.544744014199003, 5.370943413063020, 5.413154400838714, 6.011820435049564, 6.546125095704613, 7.172253721540431, 7.605451171392957, 8.182517542175901, 8.491483644249331, 9.137849938013737, 9.765957505993618, 10.27316579916380, 10.46458797534281, 11.06525475496086, 11.51353455087903, 11.98508172673626, 12.48117225425723, 12.83046251347308, 13.45747607584295
