| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s + 2·10-s − 2·13-s − 14-s + 16-s − 7·17-s + 6·19-s − 2·20-s + 8·23-s − 25-s + 2·26-s + 28-s + 7·29-s + 7·31-s − 32-s + 7·34-s − 2·35-s + 4·37-s − 6·38-s + 2·40-s − 5·41-s + 11·43-s − 8·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.69·17-s + 1.37·19-s − 0.447·20-s + 1.66·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s + 1.29·29-s + 1.25·31-s − 0.176·32-s + 1.20·34-s − 0.338·35-s + 0.657·37-s − 0.973·38-s + 0.316·40-s − 0.780·41-s + 1.67·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.632111474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632111474\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
| 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
−14.86271274124030, −14.02002240330081, −13.75763591706893, −12.99161928351615, −12.41079966785443, −11.86883572014492, −11.50850227271326, −10.94620050312448, −10.64211333956844, −9.783676721208191, −9.362197152228726, −8.823389560981316, −8.250791881712275, −7.794068683083018, −7.257375846097440, −6.748542425411210, −6.258050604889630, −5.216818767523273, −4.850157529895480, −4.188797485632056, −3.471827196463151, −2.642811387986924, −2.279938500631907, −1.024127593002399, −0.6369583574495854,
0.6369583574495854, 1.024127593002399, 2.279938500631907, 2.642811387986924, 3.471827196463151, 4.188797485632056, 4.850157529895480, 5.216818767523273, 6.258050604889630, 6.748542425411210, 7.257375846097440, 7.794068683083018, 8.250791881712275, 8.823389560981316, 9.362197152228726, 9.783676721208191, 10.64211333956844, 10.94620050312448, 11.50850227271326, 11.86883572014492, 12.41079966785443, 12.99161928351615, 13.75763591706893, 14.02002240330081, 14.86271274124030
