| L(s) = 1 | − 3-s − 7-s + 9-s + 6·13-s + 17-s − 19-s + 21-s − 4·23-s − 5·25-s − 27-s + 29-s + 8·31-s + 11·37-s − 6·39-s − 4·41-s + 43-s − 6·47-s − 6·49-s − 51-s − 4·53-s + 57-s − 3·59-s + 4·61-s − 63-s + 7·67-s + 4·69-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.242·17-s − 0.229·19-s + 0.218·21-s − 0.834·23-s − 25-s − 0.192·27-s + 0.185·29-s + 1.43·31-s + 1.80·37-s − 0.960·39-s − 0.624·41-s + 0.152·43-s − 0.875·47-s − 6/7·49-s − 0.140·51-s − 0.549·53-s + 0.132·57-s − 0.390·59-s + 0.512·61-s − 0.125·63-s + 0.855·67-s + 0.481·69-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.30186177202202, −14.43551310463831, −14.00145399317122, −13.39639730053255, −13.04844053627592, −12.50275547285544, −11.77464975518091, −11.46867290262620, −10.95792534347366, −10.35461871490535, −9.694899952073248, −9.543417234737489, −8.401503016696820, −8.276178651685368, −7.622493095597671, −6.699567301864235, −6.307408679612081, −5.958920587953737, −5.290229077373475, −4.446456576286530, −3.996069029110120, −3.341347677360281, −2.578503015347712, −1.630018261087860, −0.9911833799778977, 0,
0.9911833799778977, 1.630018261087860, 2.578503015347712, 3.341347677360281, 3.996069029110120, 4.446456576286530, 5.290229077373475, 5.958920587953737, 6.307408679612081, 6.699567301864235, 7.622493095597671, 8.276178651685368, 8.401503016696820, 9.543417234737489, 9.694899952073248, 10.35461871490535, 10.95792534347366, 11.46867290262620, 11.77464975518091, 12.50275547285544, 13.04844053627592, 13.39639730053255, 14.00145399317122, 14.43551310463831, 15.30186177202202
