| L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 11-s − 2·15-s − 6·19-s + 4·21-s − 8·23-s − 25-s − 27-s + 8·29-s − 4·31-s + 33-s − 8·35-s + 37-s − 10·41-s − 10·43-s + 2·45-s − 8·47-s + 9·49-s − 14·53-s − 2·55-s + 6·57-s − 8·61-s − 4·63-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.516·15-s − 1.37·19-s + 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.174·33-s − 1.35·35-s + 0.164·37-s − 1.56·41-s − 1.52·43-s + 0.298·45-s − 1.16·47-s + 9/7·49-s − 1.92·53-s − 0.269·55-s + 0.794·57-s − 1.02·61-s − 0.503·63-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4902838209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4902838209\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.62989097572733, −15.45837699926962, −14.50395436386533, −13.96340782124494, −13.34964973813882, −13.03555615586118, −12.40335719639879, −12.03856567763473, −11.22506585919643, −10.50533113155215, −10.01978900475321, −9.831399413090141, −9.126281154652074, −8.336084798386118, −7.820858997998836, −6.688845474813289, −6.403194973546671, −6.182598547230908, −5.275361375902123, −4.711502764629490, −3.770294305209675, −3.214707152836940, −2.244831769075350, −1.689214588722664, −0.2828633783239288,
0.2828633783239288, 1.689214588722664, 2.244831769075350, 3.214707152836940, 3.770294305209675, 4.711502764629490, 5.275361375902123, 6.182598547230908, 6.403194973546671, 6.688845474813289, 7.820858997998836, 8.336084798386118, 9.126281154652074, 9.831399413090141, 10.01978900475321, 10.50533113155215, 11.22506585919643, 12.03856567763473, 12.40335719639879, 13.03555615586118, 13.34964973813882, 13.96340782124494, 14.50395436386533, 15.45837699926962, 15.62989097572733
