| L(s) = 1 | + 3·3-s + 7-s + 6·9-s + 11-s − 5·17-s + 7·19-s + 3·21-s + 8·23-s + 9·27-s + 3·29-s + 5·31-s + 3·33-s − 37-s − 8·41-s − 10·43-s − 6·49-s − 15·51-s − 53-s + 21·57-s − 12·59-s + 5·61-s + 6·63-s + 4·67-s + 24·69-s + 7·71-s + 2·73-s + 77-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s + 0.301·11-s − 1.21·17-s + 1.60·19-s + 0.654·21-s + 1.66·23-s + 1.73·27-s + 0.557·29-s + 0.898·31-s + 0.522·33-s − 0.164·37-s − 1.24·41-s − 1.52·43-s − 6/7·49-s − 2.10·51-s − 0.137·53-s + 2.78·57-s − 1.56·59-s + 0.640·61-s + 0.755·63-s + 0.488·67-s + 2.88·69-s + 0.830·71-s + 0.234·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.296227693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.296227693\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.469356911959734249901734341214, −7.79194284255075268719094657759, −7.04874225592111134799564018269, −6.51694374645993230451285053358, −5.05919685124240264323569570014, −4.60380453183220959506825778742, −3.45275627199387351119161477100, −3.05162751303296021137708057299, −2.07100995800328393712413255294, −1.18111973814886682455897140888,
1.18111973814886682455897140888, 2.07100995800328393712413255294, 3.05162751303296021137708057299, 3.45275627199387351119161477100, 4.60380453183220959506825778742, 5.05919685124240264323569570014, 6.51694374645993230451285053358, 7.04874225592111134799564018269, 7.79194284255075268719094657759, 8.469356911959734249901734341214
