| L(s) = 1 | − 2·4-s + 2·5-s − 7-s − 3·11-s − 6·13-s + 4·16-s − 2·17-s − 4·20-s − 25-s + 2·28-s + 6·29-s + 4·31-s − 2·35-s + 37-s + 9·41-s + 4·43-s + 6·44-s + 13·47-s − 6·49-s + 12·52-s + 7·53-s − 6·55-s − 6·59-s + 4·61-s − 8·64-s − 12·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s − 0.377·7-s − 0.904·11-s − 1.66·13-s + 16-s − 0.485·17-s − 0.894·20-s − 1/5·25-s + 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.338·35-s + 0.164·37-s + 1.40·41-s + 0.609·43-s + 0.904·44-s + 1.89·47-s − 6/7·49-s + 1.66·52-s + 0.961·53-s − 0.809·55-s − 0.781·59-s + 0.512·61-s − 64-s − 1.48·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120213 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120213 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359702753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359702753\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.53933249505674, −13.11218056173648, −12.74064301409256, −12.17814984402585, −11.90924910181813, −10.94143392152752, −10.43904782565928, −10.11379385666511, −9.652855662815090, −9.255815461018226, −8.830600098545609, −8.113864354619933, −7.665446759109495, −7.220552506515407, −6.429692642202727, −5.982491948743579, −5.396706706627884, −4.975830319235568, −4.474758614810156, −3.963580823159798, −3.040842778148435, −2.498958773160547, −2.186027499432045, −1.042821523973893, −0.4072557251242899,
0.4072557251242899, 1.042821523973893, 2.186027499432045, 2.498958773160547, 3.040842778148435, 3.963580823159798, 4.474758614810156, 4.975830319235568, 5.396706706627884, 5.982491948743579, 6.429692642202727, 7.220552506515407, 7.665446759109495, 8.113864354619933, 8.830600098545609, 9.255815461018226, 9.652855662815090, 10.11379385666511, 10.43904782565928, 10.94143392152752, 11.90924910181813, 12.17814984402585, 12.74064301409256, 13.11218056173648, 13.53933249505674
