| L(s) = 1 | − 3-s − 5-s + 9-s − 3·11-s + 13-s + 15-s + 8·17-s + 2·19-s − 6·23-s − 4·25-s − 27-s + 29-s − 31-s + 3·33-s + 2·37-s − 39-s + 10·41-s − 45-s − 2·47-s − 8·51-s − 3·53-s + 3·55-s − 2·57-s − 9·59-s + 2·61-s − 65-s − 10·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.258·15-s + 1.94·17-s + 0.458·19-s − 1.25·23-s − 4/5·25-s − 0.192·27-s + 0.185·29-s − 0.179·31-s + 0.522·33-s + 0.328·37-s − 0.160·39-s + 1.56·41-s − 0.149·45-s − 0.291·47-s − 1.12·51-s − 0.412·53-s + 0.404·55-s − 0.264·57-s − 1.17·59-s + 0.256·61-s − 0.124·65-s − 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−13.03548724552256, −12.53135641883159, −12.15647586760833, −11.63183961643525, −11.50320648942282, −10.66842767879820, −10.35752574833522, −10.01811654673638, −9.451164611006211, −8.999514180740171, −8.130281772983803, −7.899803359619519, −7.535626033480573, −7.142367282432381, −6.178152775033347, −5.876584141409993, −5.639423146555562, −4.884876520589548, −4.444131758016152, −3.852224936497650, −3.278592701428374, −2.847561541650256, −2.006628241795327, −1.386644964297871, −0.7008525878526418, 0,
0.7008525878526418, 1.386644964297871, 2.006628241795327, 2.847561541650256, 3.278592701428374, 3.852224936497650, 4.444131758016152, 4.884876520589548, 5.639423146555562, 5.876584141409993, 6.178152775033347, 7.142367282432381, 7.535626033480573, 7.899803359619519, 8.130281772983803, 8.999514180740171, 9.451164611006211, 10.01811654673638, 10.35752574833522, 10.66842767879820, 11.50320648942282, 11.63183961643525, 12.15647586760833, 12.53135641883159, 13.03548724552256
