| L(s) = 1 | − 2-s − 3-s + 4-s + 4·5-s + 6-s − 7-s − 8-s + 9-s − 4·10-s − 12-s − 13-s + 14-s − 4·15-s + 16-s − 6·17-s − 18-s − 8·19-s + 4·20-s + 21-s + 6·23-s + 24-s + 11·25-s + 26-s − 27-s − 28-s + 6·29-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 − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s + 0.894·20-s + 0.218·21-s + 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313224099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313224099\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.22480624176243, −13.61191385803506, −13.05847687050191, −12.77403346658527, −12.34904254849762, −11.50431891224324, −10.86473862853987, −10.62030341103430, −10.25510311075595, −9.528609544352758, −9.240588040859345, −8.675356141476717, −8.313152912620560, −7.176997999800585, −6.805651630927078, −6.431017989430854, −5.989813633984428, −5.369308419961801, −4.723284481644777, −4.234862524578153, −3.063337908241815, −2.495545612482523, −1.990395719326276, −1.360604574589804, −0.4412505985642850,
0.4412505985642850, 1.360604574589804, 1.990395719326276, 2.495545612482523, 3.063337908241815, 4.234862524578153, 4.723284481644777, 5.369308419961801, 5.989813633984428, 6.431017989430854, 6.805651630927078, 7.176997999800585, 8.313152912620560, 8.675356141476717, 9.240588040859345, 9.528609544352758, 10.25510311075595, 10.62030341103430, 10.86473862853987, 11.50431891224324, 12.34904254849762, 12.77403346658527, 13.05847687050191, 13.61191385803506, 14.22480624176243
