| L(s) = 1 | + 3-s + 7-s + 9-s − 13-s + 17-s + 8·19-s + 21-s + 5·23-s + 27-s − 2·29-s + 5·31-s + 3·37-s − 39-s − 3·41-s + 12·43-s + 3·47-s + 49-s + 51-s + 12·53-s + 8·57-s + 12·59-s + 11·61-s + 63-s − 2·67-s + 5·69-s + 10·71-s − 8·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.242·17-s + 1.83·19-s + 0.218·21-s + 1.04·23-s + 0.192·27-s − 0.371·29-s + 0.898·31-s + 0.493·37-s − 0.160·39-s − 0.468·41-s + 1.82·43-s + 0.437·47-s + 1/7·49-s + 0.140·51-s + 1.64·53-s + 1.05·57-s + 1.56·59-s + 1.40·61-s + 0.125·63-s − 0.244·67-s + 0.601·69-s + 1.18·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.071620961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.071620961\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−14.80120331548287, −14.47571311147842, −13.84656610873337, −13.49689696084687, −12.88139596884976, −12.34561076421120, −11.60713962006239, −11.45528214520242, −10.61253867327357, −10.02335875934453, −9.622338462503315, −8.977305998640766, −8.546408452196938, −7.838029223374127, −7.329476940491317, −7.014748135492310, −6.105402797518533, −5.324343699820743, −5.088403172273925, −4.111040262013181, −3.686850959753963, −2.728144076806841, −2.490155103477837, −1.298858947676682, −0.8231289185156980,
0.8231289185156980, 1.298858947676682, 2.490155103477837, 2.728144076806841, 3.686850959753963, 4.111040262013181, 5.088403172273925, 5.324343699820743, 6.105402797518533, 7.014748135492310, 7.329476940491317, 7.838029223374127, 8.546408452196938, 8.977305998640766, 9.622338462503315, 10.02335875934453, 10.61253867327357, 11.45528214520242, 11.60713962006239, 12.34561076421120, 12.88139596884976, 13.49689696084687, 13.84656610873337, 14.47571311147842, 14.80120331548287
