| L(s) = 1 | − 7-s − 6·13-s + 17-s − 8·19-s + 6·29-s − 2·31-s − 6·37-s − 8·41-s − 10·43-s + 12·47-s + 49-s + 12·53-s + 4·59-s − 12·61-s − 14·67-s + 4·71-s − 14·73-s + 4·79-s − 4·83-s − 10·89-s + 6·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.66·13-s + 0.242·17-s − 1.83·19-s + 1.11·29-s − 0.359·31-s − 0.986·37-s − 1.24·41-s − 1.52·43-s + 1.75·47-s + 1/7·49-s + 1.64·53-s + 0.520·59-s − 1.53·61-s − 1.71·67-s + 0.474·71-s − 1.63·73-s + 0.450·79-s − 0.439·83-s − 1.05·89-s + 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.49854684949674, −12.94076187946823, −12.44463494997492, −12.04179246175303, −11.88348382731655, −11.08652898590353, −10.41318471596307, −10.22843043220390, −9.961805777231031, −9.106849484376450, −8.750172847461436, −8.427332899479723, −7.671366360918327, −7.212355619554074, −6.850175220443442, −6.318044177191354, −5.765035733611880, −5.172655677357610, −4.674553037737302, −4.256669160707586, −3.588190986286071, −2.960266124686722, −2.418946594201806, −1.953551474337780, −1.172452812610621, 0, 0,
1.172452812610621, 1.953551474337780, 2.418946594201806, 2.960266124686722, 3.588190986286071, 4.256669160707586, 4.674553037737302, 5.172655677357610, 5.765035733611880, 6.318044177191354, 6.850175220443442, 7.212355619554074, 7.671366360918327, 8.427332899479723, 8.750172847461436, 9.106849484376450, 9.961805777231031, 10.22843043220390, 10.41318471596307, 11.08652898590353, 11.88348382731655, 12.04179246175303, 12.44463494997492, 12.94076187946823, 13.49854684949674
