| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s + 2·11-s + 12-s + 2·13-s + 16-s + 4·17-s + 2·18-s + 6·19-s − 2·22-s + 7·23-s − 24-s − 2·26-s − 5·27-s + 5·29-s − 6·31-s − 32-s + 2·33-s − 4·34-s − 2·36-s + 2·37-s − 6·38-s + 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.603·11-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.970·17-s + 0.471·18-s + 1.37·19-s − 0.426·22-s + 1.45·23-s − 0.204·24-s − 0.392·26-s − 0.962·27-s + 0.928·29-s − 1.07·31-s − 0.176·32-s + 0.348·33-s − 0.685·34-s − 1/3·36-s + 0.328·37-s − 0.973·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.230895643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.230895643\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.07440709034728, −12.65582256150929, −11.96895062407311, −11.57534026015120, −11.30039413658960, −10.69732601152021, −10.21917990301369, −9.525605076449459, −9.346818739110885, −8.880934602203241, −8.338907344589269, −7.973373966016065, −7.437109381381173, −6.945233631404820, −6.486433239224263, −5.719154323687470, −5.455574711590310, −4.843672899485176, −3.923915594736785, −3.473808890150761, −3.018801647798437, −2.533750786096240, −1.718017347518153, −1.070797638356916, −0.6429809001083153,
0.6429809001083153, 1.070797638356916, 1.718017347518153, 2.533750786096240, 3.018801647798437, 3.473808890150761, 3.923915594736785, 4.843672899485176, 5.455574711590310, 5.719154323687470, 6.486433239224263, 6.945233631404820, 7.437109381381173, 7.973373966016065, 8.338907344589269, 8.880934602203241, 9.346818739110885, 9.525605076449459, 10.21917990301369, 10.69732601152021, 11.30039413658960, 11.57534026015120, 11.96895062407311, 12.65582256150929, 13.07440709034728
