| L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 3·7-s − 8-s + 9-s − 10-s − 4·11-s − 2·12-s − 5·13-s + 3·14-s − 2·15-s + 16-s + 17-s − 18-s + 4·19-s + 20-s + 6·21-s + 4·22-s + 2·23-s + 2·24-s + 25-s + 5·26-s + 4·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.577·12-s − 1.38·13-s + 0.801·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 1.30·21-s + 0.852·22-s + 0.417·23-s + 0.408·24-s + 1/5·25-s + 0.980·26-s + 0.769·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.14689089322206, −12.91683338040735, −12.26300479403115, −11.86871333259935, −11.72379824756486, −10.89100449598862, −10.44880093663169, −10.09111196302098, −9.886558696022225, −9.408057566782730, −8.597109365337508, −8.394237777931165, −7.561321801184535, −7.055200905412966, −6.872781554475237, −6.260056755441048, −5.683175754788897, −5.296057402694531, −4.977773868934858, −4.302318070399941, −3.227386150123176, −2.915647562163781, −2.526373497453881, −1.606728656601707, −0.9162575606294470, 0, 0,
0.9162575606294470, 1.606728656601707, 2.526373497453881, 2.915647562163781, 3.227386150123176, 4.302318070399941, 4.977773868934858, 5.296057402694531, 5.683175754788897, 6.260056755441048, 6.872781554475237, 7.055200905412966, 7.561321801184535, 8.394237777931165, 8.597109365337508, 9.408057566782730, 9.886558696022225, 10.09111196302098, 10.44880093663169, 10.89100449598862, 11.72379824756486, 11.86871333259935, 12.26300479403115, 12.91683338040735, 13.14689089322206
