| L(s) = 1 | − 7-s − 6·11-s − 13-s − 3·17-s − 6·19-s + 3·23-s + 6·29-s + 6·31-s − 2·37-s + 47-s + 49-s − 4·53-s + 59-s − 14·61-s − 5·67-s − 6·71-s + 5·73-s + 6·77-s − 4·79-s − 4·83-s − 12·89-s + 91-s + 9·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.80·11-s − 0.277·13-s − 0.727·17-s − 1.37·19-s + 0.625·23-s + 1.11·29-s + 1.07·31-s − 0.328·37-s + 0.145·47-s + 1/7·49-s − 0.549·53-s + 0.130·59-s − 1.79·61-s − 0.610·67-s − 0.712·71-s + 0.585·73-s + 0.683·77-s − 0.450·79-s − 0.439·83-s − 1.27·89-s + 0.104·91-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2675020419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2675020419\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−12.71022253899695, −12.19372010547994, −11.80905387641755, −11.03467953010495, −10.75102229963810, −10.40871114359065, −9.977312458412207, −9.478771932833659, −8.762451723205729, −8.553698570610837, −8.022522148992465, −7.554631269854631, −7.026371885304107, −6.533039287714887, −6.090295257194266, −5.600620953923617, −4.844968979396392, −4.681905537541368, −4.163242258030582, −3.283188468454457, −2.824204455519759, −2.478516379045286, −1.877002577873185, −1.030288235356974, −0.1441485502601130,
0.1441485502601130, 1.030288235356974, 1.877002577873185, 2.478516379045286, 2.824204455519759, 3.283188468454457, 4.163242258030582, 4.681905537541368, 4.844968979396392, 5.600620953923617, 6.090295257194266, 6.533039287714887, 7.026371885304107, 7.554631269854631, 8.022522148992465, 8.553698570610837, 8.762451723205729, 9.478771932833659, 9.977312458412207, 10.40871114359065, 10.75102229963810, 11.03467953010495, 11.80905387641755, 12.19372010547994, 12.71022253899695
