| L(s) = 1 | + 3-s − 3·5-s − 4·7-s + 9-s + 2·11-s − 2·13-s − 3·15-s + 4·17-s + 19-s − 4·21-s − 6·23-s + 4·25-s + 27-s + 8·29-s + 2·33-s + 12·35-s − 7·37-s − 2·39-s + 41-s + 43-s − 3·45-s + 2·47-s + 9·49-s + 4·51-s − 6·53-s − 6·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.774·15-s + 0.970·17-s + 0.229·19-s − 0.872·21-s − 1.25·23-s + 4/5·25-s + 0.192·27-s + 1.48·29-s + 0.348·33-s + 2.02·35-s − 1.15·37-s − 0.320·39-s + 0.156·41-s + 0.152·43-s − 0.447·45-s + 0.291·47-s + 9/7·49-s + 0.560·51-s − 0.824·53-s − 0.809·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−15.24353655791138, −14.41465198902547, −14.12453582822858, −13.65109463853260, −12.76038452105582, −12.46630090741744, −11.98878357344746, −11.71352274492523, −10.78206225945219, −10.10507134170432, −9.905050812780470, −9.183225659052001, −8.699663688333266, −8.056957558216255, −7.519639208442103, −7.168341938335698, −6.360183524533686, −6.053310298835516, −5.003424090742550, −4.358714481637857, −3.760905848226125, −3.264965795624453, −2.896797525287513, −1.862751792317149, −0.8040495695756988, 0,
0.8040495695756988, 1.862751792317149, 2.896797525287513, 3.264965795624453, 3.760905848226125, 4.358714481637857, 5.003424090742550, 6.053310298835516, 6.360183524533686, 7.168341938335698, 7.519639208442103, 8.056957558216255, 8.699663688333266, 9.183225659052001, 9.905050812780470, 10.10507134170432, 10.78206225945219, 11.71352274492523, 11.98878357344746, 12.46630090741744, 12.76038452105582, 13.65109463853260, 14.12453582822858, 14.41465198902547, 15.24353655791138
