| L(s) = 1 | + 5-s + 6·13-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 6·29-s − 6·37-s − 2·41-s + 4·43-s + 8·47-s − 2·53-s + 12·59-s + 6·61-s + 6·65-s + 4·67-s + 12·71-s − 10·73-s − 8·79-s + 12·83-s − 2·85-s + 14·89-s + 4·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.66·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.986·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 0.274·53-s + 1.56·59-s + 0.768·61-s + 0.744·65-s + 0.488·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s + 1.31·83-s − 0.216·85-s + 1.48·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.645059184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.645059184\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−13.38829497698747, −13.11948267940779, −12.39791662586909, −11.95719109440577, −11.51871639301206, −10.94859676471671, −10.52221796452478, −10.10801865958759, −9.530273152723993, −8.994420364894827, −8.552682384493418, −8.183127124026152, −7.524634023257588, −6.914949949615620, −6.408722638322814, −6.054384376301100, −5.360004550883497, −5.073636058988713, −4.088520568801257, −3.869454763453953, −3.163638782366828, −2.510791036611555, −1.876967451797419, −1.182828143743836, −0.6214580226434191,
0.6214580226434191, 1.182828143743836, 1.876967451797419, 2.510791036611555, 3.163638782366828, 3.869454763453953, 4.088520568801257, 5.073636058988713, 5.360004550883497, 6.054384376301100, 6.408722638322814, 6.914949949615620, 7.524634023257588, 8.183127124026152, 8.552682384493418, 8.994420364894827, 9.530273152723993, 10.10801865958759, 10.52221796452478, 10.94859676471671, 11.51871639301206, 11.95719109440577, 12.39791662586909, 13.11948267940779, 13.38829497698747
