| L(s) = 1 | + 2·5-s + 4·7-s − 2·13-s + 2·17-s + 4·19-s − 23-s − 25-s + 2·29-s + 8·31-s + 8·35-s + 2·37-s − 10·41-s + 4·43-s + 9·49-s − 6·53-s + 12·59-s + 2·61-s − 4·65-s + 12·67-s − 16·71-s + 10·73-s + 4·79-s + 4·85-s − 6·89-s − 8·91-s + 8·95-s − 14·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s − 1/5·25-s + 0.371·29-s + 1.43·31-s + 1.35·35-s + 0.328·37-s − 1.56·41-s + 0.609·43-s + 9/7·49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s − 0.496·65-s + 1.46·67-s − 1.89·71-s + 1.17·73-s + 0.450·79-s + 0.433·85-s − 0.635·89-s − 0.838·91-s + 0.820·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.824987577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.824987577\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 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.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.449404081151448975498660149604, −8.015246531075270933859954344414, −7.21416507761077576878720371711, −6.33418191884306148257172980045, −5.38896446077744874211346230959, −5.02927292118075749716492845672, −4.08919503517483251315083737718, −2.84258678708212489571040658128, −1.96184092082491690327825150977, −1.09797971434280589622224992157,
1.09797971434280589622224992157, 1.96184092082491690327825150977, 2.84258678708212489571040658128, 4.08919503517483251315083737718, 5.02927292118075749716492845672, 5.38896446077744874211346230959, 6.33418191884306148257172980045, 7.21416507761077576878720371711, 8.015246531075270933859954344414, 8.449404081151448975498660149604
