| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 6·11-s + 12-s − 4·13-s + 16-s + 18-s − 19-s + 6·22-s − 6·23-s + 24-s − 4·26-s + 27-s + 6·29-s + 4·31-s + 32-s + 6·33-s + 36-s − 2·37-s − 38-s − 4·39-s + 6·41-s − 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.235·18-s − 0.229·19-s + 1.27·22-s − 1.25·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.04·33-s + 1/6·36-s − 0.328·37-s − 0.162·38-s − 0.640·39-s + 0.937·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.182544830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.182544830\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.62245866606917, −12.91987722706245, −12.36257484248367, −12.09141783699319, −11.67245120362964, −11.26141836394368, −10.28238513909923, −10.16418927220616, −9.603949971677890, −8.996499912396759, −8.576187934287295, −8.056848309279063, −7.335649919852119, −7.056137158648326, −6.399692776664728, −6.094236782970938, −5.372631324844952, −4.687695066679124, −4.163702817273432, −3.969276450689938, −3.143818185683066, −2.641042857697052, −2.007308500576579, −1.429715973239468, −0.6244741387759008,
0.6244741387759008, 1.429715973239468, 2.007308500576579, 2.641042857697052, 3.143818185683066, 3.969276450689938, 4.163702817273432, 4.687695066679124, 5.372631324844952, 6.094236782970938, 6.399692776664728, 7.056137158648326, 7.335649919852119, 8.056848309279063, 8.576187934287295, 8.996499912396759, 9.603949971677890, 10.16418927220616, 10.28238513909923, 11.26141836394368, 11.67245120362964, 12.09141783699319, 12.36257484248367, 12.91987722706245, 13.62245866606917
