| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 13-s − 15-s − 5·17-s + 8·19-s + 2·21-s + 6·23-s − 4·25-s + 27-s − 3·29-s + 4·31-s − 2·35-s − 11·37-s − 39-s + 3·41-s − 2·43-s − 45-s + 4·47-s − 3·49-s − 5·51-s + 3·53-s + 8·57-s + 6·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s − 1.21·17-s + 1.83·19-s + 0.436·21-s + 1.25·23-s − 4/5·25-s + 0.192·27-s − 0.557·29-s + 0.718·31-s − 0.338·35-s − 1.80·37-s − 0.160·39-s + 0.468·41-s − 0.304·43-s − 0.149·45-s + 0.583·47-s − 3/7·49-s − 0.700·51-s + 0.412·53-s + 1.05·57-s + 0.781·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.677624979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.677624979\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−16.25373587616659, −15.66167868152343, −15.40115380333986, −14.69657214024130, −14.20203940987783, −13.54289522107157, −13.24236719439202, −12.32674605162263, −11.77232248273527, −11.31372031412399, −10.72287678013556, −9.930113522055265, −9.346013371008182, −8.730945400940516, −8.220772242421927, −7.414837750267667, −7.196684352440457, −6.312400673402291, −5.171434647833054, −4.991265339972027, −3.981687651362406, −3.414656624561270, −2.531242104817478, −1.757140616114861, −0.7409349410587745,
0.7409349410587745, 1.757140616114861, 2.531242104817478, 3.414656624561270, 3.981687651362406, 4.991265339972027, 5.171434647833054, 6.312400673402291, 7.196684352440457, 7.414837750267667, 8.220772242421927, 8.730945400940516, 9.346013371008182, 9.930113522055265, 10.72287678013556, 11.31372031412399, 11.77232248273527, 12.32674605162263, 13.24236719439202, 13.54289522107157, 14.20203940987783, 14.69657214024130, 15.40115380333986, 15.66167868152343, 16.25373587616659
