| L(s) = 1 | − 2-s + 4-s + 4·5-s + 7-s − 8-s − 4·10-s − 6·13-s − 14-s + 16-s + 4·19-s + 4·20-s + 6·23-s + 11·25-s + 6·26-s + 28-s + 4·29-s + 4·31-s − 32-s + 4·35-s + 6·37-s − 4·38-s − 4·40-s − 2·41-s − 5·43-s − 6·46-s + 13·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s − 0.353·8-s − 1.26·10-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.917·19-s + 0.894·20-s + 1.25·23-s + 11/5·25-s + 1.17·26-s + 0.188·28-s + 0.742·29-s + 0.718·31-s − 0.176·32-s + 0.676·35-s + 0.986·37-s − 0.648·38-s − 0.632·40-s − 0.312·41-s − 0.762·43-s − 0.884·46-s + 1.89·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.012270200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.012270200\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−14.61588178976803, −14.07746957252686, −13.69291523123116, −13.16975093187257, −12.38266558830767, −12.21374549761081, −11.34940837019114, −10.90599498834958, −10.13071924821867, −9.960018930277584, −9.498616972591477, −8.963808392904122, −8.459828156635606, −7.660564539460835, −7.133414188803947, −6.719667391148492, −5.996011188910830, −5.427841705543004, −4.995057016121151, −4.401289726999015, −3.084906792645576, −2.650590424573943, −2.148355656974421, −1.337071403041124, −0.7359786400500553,
0.7359786400500553, 1.337071403041124, 2.148355656974421, 2.650590424573943, 3.084906792645576, 4.401289726999015, 4.995057016121151, 5.427841705543004, 5.996011188910830, 6.719667391148492, 7.133414188803947, 7.660564539460835, 8.459828156635606, 8.963808392904122, 9.498616972591477, 9.960018930277584, 10.13071924821867, 10.90599498834958, 11.34940837019114, 12.21374549761081, 12.38266558830767, 13.16975093187257, 13.69291523123116, 14.07746957252686, 14.61588178976803
