| L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s + 2·7-s + 8-s + 9-s + 4·10-s − 11-s − 12-s + 4·13-s + 2·14-s − 4·15-s + 16-s + 18-s + 4·20-s − 2·21-s − 22-s + 6·23-s − 24-s + 11·25-s + 4·26-s − 27-s + 2·28-s − 10·29-s − 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.301·11-s − 0.288·12-s + 1.10·13-s + 0.534·14-s − 1.03·15-s + 1/4·16-s + 0.235·18-s + 0.894·20-s − 0.436·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 11/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.85·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.780180882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.780180882\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.63639895289975, −15.06605771716881, −14.54372052409317, −14.01179103012109, −13.40647273731484, −13.09965213164506, −12.75536317332851, −11.70083776942865, −11.40050179397304, −10.64265750601040, −10.45969460710528, −9.621690813292869, −9.088326153210178, −8.429499588932859, −7.605627291656647, −6.892893666909340, −6.246299441046354, −5.872114184583398, −5.234274335940401, −4.880665533574897, −4.019643964500830, −3.093235120498489, −2.322973150328417, −1.603583578616473, −1.015472521954458,
1.015472521954458, 1.603583578616473, 2.322973150328417, 3.093235120498489, 4.019643964500830, 4.880665533574897, 5.234274335940401, 5.872114184583398, 6.246299441046354, 6.892893666909340, 7.605627291656647, 8.429499588932859, 9.088326153210178, 9.621690813292869, 10.45969460710528, 10.64265750601040, 11.40050179397304, 11.70083776942865, 12.75536317332851, 13.09965213164506, 13.40647273731484, 14.01179103012109, 14.54372052409317, 15.06605771716881, 15.63639895289975
