| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 2·13-s + 15-s − 4·17-s + 4·19-s − 4·21-s − 2·23-s + 25-s − 27-s − 10·29-s + 4·31-s − 4·35-s − 8·37-s + 2·39-s − 10·41-s − 12·43-s − 45-s + 8·47-s + 9·49-s + 4·51-s − 4·53-s − 4·57-s + 12·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.970·17-s + 0.917·19-s − 0.872·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.676·35-s − 1.31·37-s + 0.320·39-s − 1.56·41-s − 1.82·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.560·51-s − 0.549·53-s − 0.529·57-s + 1.56·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 79 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.16809846840478, −14.63940519743283, −14.11029065034104, −13.55578492039808, −13.06590855362607, −12.26544454555891, −11.85734679081138, −11.46299643275701, −11.08321512020377, −10.48773639090878, −9.891900946739121, −9.286046424745577, −8.555729706597619, −8.089613633650640, −7.653123679316366, −6.856633402644005, −6.667504250367962, −5.486762888868776, −5.131096478168485, −4.844368223701222, −3.882283868423870, −3.549702238951251, −2.237063695437587, −1.898818784904593, −0.9462745352325097, 0,
0.9462745352325097, 1.898818784904593, 2.237063695437587, 3.549702238951251, 3.882283868423870, 4.844368223701222, 5.131096478168485, 5.486762888868776, 6.667504250367962, 6.856633402644005, 7.653123679316366, 8.089613633650640, 8.555729706597619, 9.286046424745577, 9.891900946739121, 10.48773639090878, 11.08321512020377, 11.46299643275701, 11.85734679081138, 12.26544454555891, 13.06590855362607, 13.55578492039808, 14.11029065034104, 14.63940519743283, 15.16809846840478
