| L(s) = 1 | + 3·3-s − 7-s + 6·9-s − 13-s + 5·17-s + 8·19-s − 3·21-s + 2·23-s + 9·27-s + 29-s − 2·31-s + 10·37-s − 3·39-s + 6·41-s + 4·43-s + 11·47-s + 49-s + 15·51-s + 6·53-s + 24·57-s − 10·59-s − 6·63-s − 10·67-s + 6·69-s + 10·73-s + 7·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s − 0.277·13-s + 1.21·17-s + 1.83·19-s − 0.654·21-s + 0.417·23-s + 1.73·27-s + 0.185·29-s − 0.359·31-s + 1.64·37-s − 0.480·39-s + 0.937·41-s + 0.609·43-s + 1.60·47-s + 1/7·49-s + 2.10·51-s + 0.824·53-s + 3.17·57-s − 1.30·59-s − 0.755·63-s − 1.22·67-s + 0.722·69-s + 1.17·73-s + 0.787·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.898558430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.898558430\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−13.91577853568572, −13.66484166040541, −13.04138872627378, −12.55720518794550, −12.13255857274509, −11.55237645129165, −10.80921111374083, −10.24377518310321, −9.739235661389362, −9.265714151121719, −9.152676711302840, −8.362157283035399, −7.702532817097997, −7.529672699368727, −7.165416632324067, −6.219852085935306, −5.680140878401911, −5.032523859050165, −4.288047347043166, −3.758987751916099, −3.179604937596137, −2.778099942930972, −2.270332157387352, −1.274442541040385, −0.8467429321915256,
0.8467429321915256, 1.274442541040385, 2.270332157387352, 2.778099942930972, 3.179604937596137, 3.758987751916099, 4.288047347043166, 5.032523859050165, 5.680140878401911, 6.219852085935306, 7.165416632324067, 7.529672699368727, 7.702532817097997, 8.362157283035399, 9.152676711302840, 9.265714151121719, 9.739235661389362, 10.24377518310321, 10.80921111374083, 11.55237645129165, 12.13255857274509, 12.55720518794550, 13.04138872627378, 13.66484166040541, 13.91577853568572
