| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 3·11-s + 12-s − 15-s + 16-s − 3·17-s − 18-s − 19-s − 20-s − 3·22-s − 6·23-s − 24-s + 25-s + 27-s − 3·29-s + 30-s − 8·31-s − 32-s + 3·33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.639·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.557·29-s + 0.182·30-s − 1.43·31-s − 0.176·32-s + 0.522·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.92153773650023, −12.71210780803504, −12.05822812160268, −11.60068462924419, −11.25726262215324, −10.76990871990233, −10.25124074497539, −9.693876545181262, −9.275091334217578, −8.986332872421110, −8.412450884727083, −7.946025329050881, −7.646240159789381, −6.895505953553717, −6.741332765586978, −6.015539791177450, −5.597290267481967, −4.776860384242395, −4.064002678099333, −3.971484493483150, −3.200600421365215, −2.662046177565016, −1.809222244304115, −1.719221766846456, −0.7034614961356102, 0,
0.7034614961356102, 1.719221766846456, 1.809222244304115, 2.662046177565016, 3.200600421365215, 3.971484493483150, 4.064002678099333, 4.776860384242395, 5.597290267481967, 6.015539791177450, 6.741332765586978, 6.895505953553717, 7.646240159789381, 7.946025329050881, 8.412450884727083, 8.986332872421110, 9.275091334217578, 9.693876545181262, 10.25124074497539, 10.76990871990233, 11.25726262215324, 11.60068462924419, 12.05822812160268, 12.71210780803504, 12.92153773650023
