| L(s) = 1 | − 2-s + 4-s + 5-s + 3·7-s − 8-s − 10-s − 11-s + 2·13-s − 3·14-s + 16-s − 8·17-s + 4·19-s + 20-s + 22-s − 6·23-s + 25-s − 2·26-s + 3·28-s − 2·29-s − 7·31-s − 32-s + 8·34-s + 3·35-s + 10·37-s − 4·38-s − 40-s − 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.554·13-s − 0.801·14-s + 1/4·16-s − 1.94·17-s + 0.917·19-s + 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s − 0.392·26-s + 0.566·28-s − 0.371·29-s − 1.25·31-s − 0.176·32-s + 1.37·34-s + 0.507·35-s + 1.64·37-s − 0.648·38-s − 0.158·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.701197595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.701197595\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 263 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 17 T + p T^{2} \) | 1.89.ar |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.49580051186756, −14.87427407352857, −14.55833709871243, −13.68975759072900, −13.39698879925986, −12.85639307427693, −11.87936996271918, −11.47830047426297, −11.17575458856966, −10.39753659941750, −10.06877153223267, −9.270141072157397, −8.661871434135977, −8.457018404391522, −7.600899416746623, −7.175597127207453, −6.487071180720912, −5.687117921763152, −5.352800944524097, −4.377912257423193, −3.911303304941278, −2.786297017461432, −2.069875270577684, −1.630738283012276, −0.5781456593998382,
0.5781456593998382, 1.630738283012276, 2.069875270577684, 2.786297017461432, 3.911303304941278, 4.377912257423193, 5.352800944524097, 5.687117921763152, 6.487071180720912, 7.175597127207453, 7.600899416746623, 8.457018404391522, 8.661871434135977, 9.270141072157397, 10.06877153223267, 10.39753659941750, 11.17575458856966, 11.47830047426297, 11.87936996271918, 12.85639307427693, 13.39698879925986, 13.68975759072900, 14.55833709871243, 14.87427407352857, 15.49580051186756
