| L(s) = 1 | + 2-s − 4-s − 3·8-s − 3·13-s − 16-s − 2·17-s − 19-s + 2·23-s − 3·26-s + 8·29-s + 8·31-s + 5·32-s − 2·34-s − 7·37-s − 38-s + 8·43-s + 2·46-s + 10·47-s + 3·52-s − 14·53-s + 8·58-s + 10·59-s − 7·61-s + 8·62-s + 7·64-s + 5·67-s + 2·68-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.832·13-s − 1/4·16-s − 0.485·17-s − 0.229·19-s + 0.417·23-s − 0.588·26-s + 1.48·29-s + 1.43·31-s + 0.883·32-s − 0.342·34-s − 1.15·37-s − 0.162·38-s + 1.21·43-s + 0.294·46-s + 1.45·47-s + 0.416·52-s − 1.92·53-s + 1.05·58-s + 1.30·59-s − 0.896·61-s + 1.01·62-s + 7/8·64-s + 0.610·67-s + 0.242·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−16.98980943158564, −15.78171192128988, −15.73845708767162, −15.00804655684081, −14.19002893858908, −14.13338440099062, −13.40140369694219, −12.77120797630106, −12.25842770183331, −11.89547843260915, −11.03880774524251, −10.36528198620419, −9.766680418862057, −9.152663896285236, −8.530898228155789, −7.996226104040380, −7.036288899380108, −6.522517440940213, −5.758821893702990, −5.066203640365130, −4.510923826530731, −3.980237636677620, −2.930977779871187, −2.511474550622957, −1.126725937862642, 0,
1.126725937862642, 2.511474550622957, 2.930977779871187, 3.980237636677620, 4.510923826530731, 5.066203640365130, 5.758821893702990, 6.522517440940213, 7.036288899380108, 7.996226104040380, 8.530898228155789, 9.152663896285236, 9.766680418862057, 10.36528198620419, 11.03880774524251, 11.89547843260915, 12.25842770183331, 12.77120797630106, 13.40140369694219, 14.13338440099062, 14.19002893858908, 15.00804655684081, 15.73845708767162, 15.78171192128988, 16.98980943158564
