| L(s) = 1 | − 4·5-s + 2·7-s + 2·11-s − 6·17-s + 2·19-s + 8·23-s + 11·25-s − 6·29-s − 10·31-s − 8·35-s + 4·37-s + 4·43-s − 2·47-s − 3·49-s + 10·53-s − 8·55-s − 14·59-s − 2·61-s + 2·67-s − 6·71-s + 8·73-s + 4·77-s + 6·83-s + 24·85-s − 8·95-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s + 0.603·11-s − 1.45·17-s + 0.458·19-s + 1.66·23-s + 11/5·25-s − 1.11·29-s − 1.79·31-s − 1.35·35-s + 0.657·37-s + 0.609·43-s − 0.291·47-s − 3/7·49-s + 1.37·53-s − 1.07·55-s − 1.82·59-s − 0.256·61-s + 0.244·67-s − 0.712·71-s + 0.936·73-s + 0.455·77-s + 0.658·83-s + 2.60·85-s − 0.820·95-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−16.54978645152185, −16.09318325155085, −15.30206221475231, −15.03170568066920, −14.67354326898293, −13.87174507180603, −13.05769140456984, −12.66913300181750, −11.90751811311233, −11.39310505121316, −11.01477505811935, −10.74309622602674, −9.327836817294419, −9.072030704593218, −8.414901307000829, −7.727175824551424, −7.233159668573239, −6.823005484074539, −5.775961982964635, −4.873709564862152, −4.462689479088696, −3.749504381513336, −3.172725374930347, −2.066935556534262, −1.039082406148430, 0,
1.039082406148430, 2.066935556534262, 3.172725374930347, 3.749504381513336, 4.462689479088696, 4.873709564862152, 5.775961982964635, 6.823005484074539, 7.233159668573239, 7.727175824551424, 8.414901307000829, 9.072030704593218, 9.327836817294419, 10.74309622602674, 11.01477505811935, 11.39310505121316, 11.90751811311233, 12.66913300181750, 13.05769140456984, 13.87174507180603, 14.67354326898293, 15.03170568066920, 15.30206221475231, 16.09318325155085, 16.54978645152185
