| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s − 2·9-s + 2·12-s − 13-s − 4·16-s − 7·17-s − 4·18-s − 6·23-s − 2·26-s − 5·27-s + 5·29-s − 2·31-s − 8·32-s − 14·34-s − 4·36-s − 2·37-s − 39-s + 2·41-s − 4·43-s − 12·46-s − 3·47-s − 4·48-s − 7·51-s − 2·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 2/3·9-s + 0.577·12-s − 0.277·13-s − 16-s − 1.69·17-s − 0.942·18-s − 1.25·23-s − 0.392·26-s − 0.962·27-s + 0.928·29-s − 0.359·31-s − 1.41·32-s − 2.40·34-s − 2/3·36-s − 0.328·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s − 1.76·46-s − 0.437·47-s − 0.577·48-s − 0.980·51-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.196367494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196367494\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.43304740467611, −13.11337959471834, −12.39005292080285, −12.09966846619552, −11.63681391910170, −11.08355944292408, −10.72533129791098, −10.01884285394458, −9.386104121168873, −8.957349642195476, −8.576700128468307, −7.940972793250250, −7.478526233127634, −6.685589148833693, −6.347809108624594, −5.934207862303442, −5.269991507345697, −4.737716876906479, −4.308070418309784, −3.814397161306907, −3.100181041733829, −2.792583323819632, −2.127213292445749, −1.657180245945892, −0.2149973318117642,
0.2149973318117642, 1.657180245945892, 2.127213292445749, 2.792583323819632, 3.100181041733829, 3.814397161306907, 4.308070418309784, 4.737716876906479, 5.269991507345697, 5.934207862303442, 6.347809108624594, 6.685589148833693, 7.478526233127634, 7.940972793250250, 8.576700128468307, 8.957349642195476, 9.386104121168873, 10.01884285394458, 10.72533129791098, 11.08355944292408, 11.63681391910170, 12.09966846619552, 12.39005292080285, 13.11337959471834, 13.43304740467611
