| L(s) = 1 | + 2-s + 4-s + 4·5-s + 8-s + 4·10-s + 11-s − 13-s + 16-s + 4·17-s − 2·19-s + 4·20-s + 22-s + 7·23-s + 11·25-s − 26-s + 8·29-s − 3·31-s + 32-s + 4·34-s + 7·37-s − 2·38-s + 4·40-s − 7·41-s − 8·43-s + 44-s + 7·46-s + 3·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.353·8-s + 1.26·10-s + 0.301·11-s − 0.277·13-s + 1/4·16-s + 0.970·17-s − 0.458·19-s + 0.894·20-s + 0.213·22-s + 1.45·23-s + 11/5·25-s − 0.196·26-s + 1.48·29-s − 0.538·31-s + 0.176·32-s + 0.685·34-s + 1.15·37-s − 0.324·38-s + 0.632·40-s − 1.09·41-s − 1.21·43-s + 0.150·44-s + 1.03·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.804042472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.804042472\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−16.66249126863539, −15.84417314873675, −15.05064535328286, −14.53133046026651, −14.25217717387362, −13.53741329694257, −13.11050810249946, −12.65389791210876, −11.98475620428804, −11.26380357208508, −10.60606384180895, −9.935217257485904, −9.729686160837970, −8.803598414917878, −8.315311457660146, −7.153004963590537, −6.774992123362750, −6.086251519640410, −5.498314271952024, −5.011347869741459, −4.269398280875196, −3.112696404883249, −2.704614905767584, −1.747952535766771, −1.085286482799235,
1.085286482799235, 1.747952535766771, 2.704614905767584, 3.112696404883249, 4.269398280875196, 5.011347869741459, 5.498314271952024, 6.086251519640410, 6.774992123362750, 7.153004963590537, 8.315311457660146, 8.803598414917878, 9.729686160837970, 9.935217257485904, 10.60606384180895, 11.26380357208508, 11.98475620428804, 12.65389791210876, 13.11050810249946, 13.53741329694257, 14.25217717387362, 14.53133046026651, 15.05064535328286, 15.84417314873675, 16.66249126863539
