| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s − 2·9-s + 10-s + 12-s − 2·13-s − 14-s − 15-s + 16-s + 3·17-s + 2·18-s + 19-s − 20-s + 21-s + 6·23-s − 24-s + 25-s + 2·26-s − 5·27-s + 28-s + 9·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.229·19-s − 0.223·20-s + 0.218·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.962·27-s + 0.188·28-s + 1.67·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.310852632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310852632\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−9.651844634834260984432132709240, −8.765998389808211963831363541535, −8.219938500505668411570488798738, −7.54599145387732851661365336346, −6.67937148953544300898990016324, −5.54997230777328407538926369145, −4.55324650873999681187519768266, −3.22052869845115697606926120635, −2.51558198103978347262901162661, −0.940612870362593859078124237356,
0.940612870362593859078124237356, 2.51558198103978347262901162661, 3.22052869845115697606926120635, 4.55324650873999681187519768266, 5.54997230777328407538926369145, 6.67937148953544300898990016324, 7.54599145387732851661365336346, 8.219938500505668411570488798738, 8.765998389808211963831363541535, 9.651844634834260984432132709240
