| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 11-s + 12-s + 5·13-s − 16-s − 3·17-s + 18-s + 2·19-s − 22-s + 8·23-s + 3·24-s + 5·26-s − 27-s − 29-s + 8·31-s + 5·32-s + 33-s − 3·34-s − 36-s − 10·37-s + 2·38-s − 5·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.38·13-s − 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s − 0.213·22-s + 1.66·23-s + 0.612·24-s + 0.980·26-s − 0.192·27-s − 0.185·29-s + 1.43·31-s + 0.883·32-s + 0.174·33-s − 0.514·34-s − 1/6·36-s − 1.64·37-s + 0.324·38-s − 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.227858227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227858227\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.60451639374980, −13.30137558956712, −12.76654470056000, −12.37069001864716, −11.66508115645568, −11.48335731439847, −10.72207034773036, −10.46612884758061, −9.817614774520048, −9.015480024492092, −8.910234875254641, −8.335051962459414, −7.659415742406357, −6.961977889922804, −6.436861972384424, −6.110026027144614, −5.367444909023052, −5.017247842523971, −4.566306641065088, −3.897318772049336, −3.315262436646239, −2.928443427962152, −1.928947181557928, −1.119035056543886, −0.4915230384994216,
0.4915230384994216, 1.119035056543886, 1.928947181557928, 2.928443427962152, 3.315262436646239, 3.897318772049336, 4.566306641065088, 5.017247842523971, 5.367444909023052, 6.110026027144614, 6.436861972384424, 6.961977889922804, 7.659415742406357, 8.335051962459414, 8.910234875254641, 9.015480024492092, 9.817614774520048, 10.46612884758061, 10.72207034773036, 11.48335731439847, 11.66508115645568, 12.37069001864716, 12.76654470056000, 13.30137558956712, 13.60451639374980
