| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 4·11-s + 12-s − 3·13-s + 14-s + 15-s + 16-s + 3·17-s + 18-s − 3·19-s + 20-s + 21-s + 4·22-s + 5·23-s + 24-s + 25-s − 3·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.688·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s + 1.04·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.706903530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.706903530\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 349 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 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
−16.53750877217341, −15.96675546700858, −15.08533950799707, −14.62913109092900, −14.48971803616989, −13.81565943054314, −13.15351208425838, −12.65075662886626, −12.01516591841502, −11.54334934915968, −10.73507102885380, −10.15773414784078, −9.485280331308063, −8.910002107403060, −8.275865627039925, −7.429825290916257, −6.939981613681729, −6.267124697375605, −5.520538056878720, −4.761516651091902, −4.220850969733460, −3.377878847591719, −2.673270505818929, −1.863889546548042, −1.044651137735001,
1.044651137735001, 1.863889546548042, 2.673270505818929, 3.377878847591719, 4.220850969733460, 4.761516651091902, 5.520538056878720, 6.267124697375605, 6.939981613681729, 7.429825290916257, 8.275865627039925, 8.910002107403060, 9.485280331308063, 10.15773414784078, 10.73507102885380, 11.54334934915968, 12.01516591841502, 12.65075662886626, 13.15351208425838, 13.81565943054314, 14.48971803616989, 14.62913109092900, 15.08533950799707, 15.96675546700858, 16.53750877217341
