| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 11-s − 6·13-s + 16-s − 17-s − 4·19-s + 2·20-s − 22-s − 25-s − 6·26-s − 2·29-s + 32-s − 34-s − 10·37-s − 4·38-s + 2·40-s − 10·41-s − 4·43-s − 44-s + 4·47-s − 50-s − 6·52-s + 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.301·11-s − 1.66·13-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 1/5·25-s − 1.17·26-s − 0.371·29-s + 0.176·32-s − 0.171·34-s − 1.64·37-s − 0.648·38-s + 0.316·40-s − 1.56·41-s − 0.609·43-s − 0.150·44-s + 0.583·47-s − 0.141·50-s − 0.832·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164934 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.759963704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759963704\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.32460799602781, −12.78864499981090, −12.31153270193620, −12.06665360825886, −11.43592636248841, −10.80204812166889, −10.45531495326272, −9.870830376456817, −9.654201021415890, −8.996529387981873, −8.321288031019543, −7.989791480769254, −7.136960585479615, −6.869727366977123, −6.473332981132247, −5.662277027146104, −5.265680006929032, −5.042010409039246, −4.236467528233210, −3.820777760627915, −2.966147105954134, −2.537748294185569, −1.931783429940049, −1.588041239213915, −0.3018457310823397,
0.3018457310823397, 1.588041239213915, 1.931783429940049, 2.537748294185569, 2.966147105954134, 3.820777760627915, 4.236467528233210, 5.042010409039246, 5.265680006929032, 5.662277027146104, 6.473332981132247, 6.869727366977123, 7.136960585479615, 7.989791480769254, 8.321288031019543, 8.996529387981873, 9.654201021415890, 9.870830376456817, 10.45531495326272, 10.80204812166889, 11.43592636248841, 12.06665360825886, 12.31153270193620, 12.78864499981090, 13.32460799602781
