| L(s) = 1 | − 2·5-s + 4·7-s − 6·13-s − 17-s + 4·19-s − 25-s + 2·29-s − 8·35-s − 10·37-s − 10·41-s − 4·43-s + 4·47-s + 9·49-s + 2·53-s − 2·61-s + 12·65-s − 12·67-s + 10·73-s + 4·79-s + 12·83-s + 2·85-s + 14·89-s − 24·91-s − 8·95-s + 18·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 1.66·13-s − 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.371·29-s − 1.35·35-s − 1.64·37-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s + 0.274·53-s − 0.256·61-s + 1.48·65-s − 1.46·67-s + 1.17·73-s + 0.450·79-s + 1.31·83-s + 0.216·85-s + 1.48·89-s − 2.51·91-s − 0.820·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.008313185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008313185\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.34197156444288, −12.15337126943967, −11.72916955653573, −11.62258816166728, −10.86332534636949, −10.40611297876013, −10.15484212240792, −9.309322084820793, −9.080919647790750, −8.354676528592160, −7.974924226093031, −7.709101903084944, −7.136699067711249, −6.873078496358607, −6.096825957942310, −5.252205490941551, −4.999419522136449, −4.820404219391345, −4.015494604956163, −3.616926883613682, −2.939373058858602, −2.282063658345393, −1.785098326111261, −1.161230215204404, −0.2746059993122686,
0.2746059993122686, 1.161230215204404, 1.785098326111261, 2.282063658345393, 2.939373058858602, 3.616926883613682, 4.015494604956163, 4.820404219391345, 4.999419522136449, 5.252205490941551, 6.096825957942310, 6.873078496358607, 7.136699067711249, 7.709101903084944, 7.974924226093031, 8.354676528592160, 9.080919647790750, 9.309322084820793, 10.15484212240792, 10.40611297876013, 10.86332534636949, 11.62258816166728, 11.72916955653573, 12.15337126943967, 12.34197156444288
