| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 2·7-s + 3·8-s + 9-s − 12-s − 2·14-s − 16-s + 17-s − 18-s − 4·19-s + 2·21-s + 4·23-s + 3·24-s − 5·25-s + 27-s − 2·28-s + 2·29-s − 8·31-s − 5·32-s − 34-s − 36-s + 6·37-s + 4·38-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s − 0.534·14-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.436·21-s + 0.834·23-s + 0.612·24-s − 25-s + 0.192·27-s − 0.377·28-s + 0.371·29-s − 1.43·31-s − 0.883·32-s − 0.171·34-s − 1/6·36-s + 0.986·37-s + 0.648·38-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−7.994906781784127211990663894550, −7.28371295954340923505439883774, −6.49137796559971784581403877698, −5.39484506423417477619985206541, −4.76463136121771652076092755676, −4.05637870775430409149091543797, −3.22842894358455373457385417561, −2.01194668166699801012124240105, −1.37704394934382279833068277144, 0,
1.37704394934382279833068277144, 2.01194668166699801012124240105, 3.22842894358455373457385417561, 4.05637870775430409149091543797, 4.76463136121771652076092755676, 5.39484506423417477619985206541, 6.49137796559971784581403877698, 7.28371295954340923505439883774, 7.994906781784127211990663894550
