| L(s) = 1 | − 2·3-s − 2·5-s − 7-s + 9-s − 4·13-s + 4·15-s − 4·17-s + 2·21-s + 4·23-s − 25-s + 4·27-s + 6·29-s − 10·31-s + 2·35-s − 6·37-s + 8·39-s − 4·41-s + 12·43-s − 2·45-s + 10·47-s + 49-s + 8·51-s − 6·53-s − 2·59-s − 63-s + 8·65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 1.03·15-s − 0.970·17-s + 0.436·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s − 1.79·31-s + 0.338·35-s − 0.986·37-s + 1.28·39-s − 0.624·41-s + 1.82·43-s − 0.298·45-s + 1.45·47-s + 1/7·49-s + 1.12·51-s − 0.824·53-s − 0.260·59-s − 0.125·63-s + 0.992·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−16.66456280969726, −15.86421033988600, −15.52174150972014, −15.01821033664150, −14.18939811578403, −13.74193015171583, −12.72288571512191, −12.42180716825497, −12.05834423984293, −11.25046912616094, −10.95877966935754, −10.43299322928026, −9.591369040459138, −9.001923932714252, −8.372627751569831, −7.415284833716489, −7.141523395813279, −6.447695337822393, −5.736025929390811, −5.069357053441686, −4.541574095530320, −3.797659971719760, −2.945646206375284, −2.075447225998316, −0.7230062005080416, 0,
0.7230062005080416, 2.075447225998316, 2.945646206375284, 3.797659971719760, 4.541574095530320, 5.069357053441686, 5.736025929390811, 6.447695337822393, 7.141523395813279, 7.415284833716489, 8.372627751569831, 9.001923932714252, 9.591369040459138, 10.43299322928026, 10.95877966935754, 11.25046912616094, 12.05834423984293, 12.42180716825497, 12.72288571512191, 13.74193015171583, 14.18939811578403, 15.01821033664150, 15.52174150972014, 15.86421033988600, 16.66456280969726
