| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s − 13-s + 16-s − 2·17-s + 18-s − 4·19-s − 8·23-s + 24-s − 26-s + 27-s + 2·29-s + 8·31-s + 32-s − 2·34-s + 36-s − 2·37-s − 4·38-s − 39-s + 6·41-s − 12·43-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 1.66·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s + 1/6·36-s − 0.328·37-s − 0.648·38-s − 0.160·39-s + 0.937·41-s − 1.82·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.764139408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.764139408\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.90926536122834, −13.25079561730960, −12.99103510969984, −12.42637756402891, −11.80938328146465, −11.61732803605866, −10.85282005160728, −10.21811294817985, −10.04147851054114, −9.380543925120884, −8.639802321367833, −8.276729547254846, −7.853564811112539, −7.150646455179903, −6.619088512301930, −6.194829267425833, −5.647574410393430, −4.831679137197116, −4.400846480987982, −4.017097144802949, −3.215383250339027, −2.757886260602188, −2.034063177611919, −1.635227723672128, −0.4870354873510213,
0.4870354873510213, 1.635227723672128, 2.034063177611919, 2.757886260602188, 3.215383250339027, 4.017097144802949, 4.400846480987982, 4.831679137197116, 5.647574410393430, 6.194829267425833, 6.619088512301930, 7.150646455179903, 7.853564811112539, 8.276729547254846, 8.639802321367833, 9.380543925120884, 10.04147851054114, 10.21811294817985, 10.85282005160728, 11.61732803605866, 11.80938328146465, 12.42637756402891, 12.99103510969984, 13.25079561730960, 13.90926536122834
