| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 11-s − 12-s − 13-s + 16-s + 4·17-s + 18-s − 2·19-s + 22-s + 6·23-s − 24-s − 26-s − 27-s − 2·29-s − 10·31-s + 32-s − 33-s + 4·34-s + 36-s + 4·37-s − 2·38-s + 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.458·19-s + 0.213·22-s + 1.25·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.371·29-s − 1.79·31-s + 0.176·32-s − 0.174·33-s + 0.685·34-s + 1/6·36-s + 0.657·37-s − 0.324·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.72322884277315, −15.20556455809633, −14.75906351919054, −14.29555411819011, −13.63716425774368, −12.94223527210002, −12.68631060442351, −12.06982870674492, −11.52130309929505, −11.00072745380546, −10.52479151217567, −9.859455728704444, −9.220520803312598, −8.636556451440601, −7.663379659322597, −7.325846010098757, −6.680821548169021, −5.963752507651188, −5.536508214779054, −4.849190413238363, −4.319669945892900, −3.467852260334300, −2.973221245455618, −1.892518534141159, −1.229848446500133, 0,
1.229848446500133, 1.892518534141159, 2.973221245455618, 3.467852260334300, 4.319669945892900, 4.849190413238363, 5.536508214779054, 5.963752507651188, 6.680821548169021, 7.325846010098757, 7.663379659322597, 8.636556451440601, 9.220520803312598, 9.859455728704444, 10.52479151217567, 11.00072745380546, 11.52130309929505, 12.06982870674492, 12.68631060442351, 12.94223527210002, 13.63716425774368, 14.29555411819011, 14.75906351919054, 15.20556455809633, 15.72322884277315
