| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s + 3·11-s + 12-s − 2·14-s − 15-s + 16-s + 3·17-s − 18-s + 5·19-s − 20-s + 2·21-s − 3·22-s + 4·23-s − 24-s − 4·25-s + 27-s + 2·28-s + 30-s − 31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.14·19-s − 0.223·20-s + 0.436·21-s − 0.639·22-s + 0.834·23-s − 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.377·28-s + 0.182·30-s − 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31434 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31434 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.708181800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.708181800\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−15.18869337958818, −14.61425961979523, −14.09578236239136, −13.68138035334947, −12.96926437773677, −12.09665978608553, −11.95350529473397, −11.31232037043299, −10.86466742072897, −10.09262767405884, −9.589881495591298, −9.162548442802115, −8.507066136401784, −8.040672971877713, −7.555811811757165, −7.049121778582400, −6.448470197602363, −5.521621448341646, −5.072018227600792, −4.117521882959662, −3.626517680954883, −2.958845280057546, −2.096246887035678, −1.358881672094607, −0.7496829141223889,
0.7496829141223889, 1.358881672094607, 2.096246887035678, 2.958845280057546, 3.626517680954883, 4.117521882959662, 5.072018227600792, 5.521621448341646, 6.448470197602363, 7.049121778582400, 7.555811811757165, 8.040672971877713, 8.507066136401784, 9.162548442802115, 9.589881495591298, 10.09262767405884, 10.86466742072897, 11.31232037043299, 11.95350529473397, 12.09665978608553, 12.96926437773677, 13.68138035334947, 14.09578236239136, 14.61425961979523, 15.18869337958818
