| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s − 4·11-s + 13-s + 16-s + 4·17-s + 2·19-s − 2·20-s + 4·22-s + 8·23-s − 25-s − 26-s + 6·29-s + 4·31-s − 32-s − 4·34-s − 2·37-s − 2·38-s + 2·40-s − 4·43-s − 4·44-s − 8·46-s − 12·47-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 1.20·11-s + 0.277·13-s + 1/4·16-s + 0.970·17-s + 0.458·19-s − 0.447·20-s + 0.852·22-s + 1.66·23-s − 1/5·25-s − 0.196·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.328·37-s − 0.324·38-s + 0.316·40-s − 0.609·43-s − 0.603·44-s − 1.17·46-s − 1.75·47-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.026411959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026411959\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 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.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−16.28993897294991, −16.04263550250978, −15.33754606729125, −15.04738475060442, −14.27197356621564, −13.48716178919939, −13.01194329839553, −12.17835000536460, −11.89453166528835, −11.03739139641071, −10.78804382853390, −9.907243586677620, −9.606504628214356, −8.547787014779639, −8.180617064320379, −7.747167180777181, −6.990154917484951, −6.487682379506930, −5.344497141337892, −5.062487669256276, −3.983665932993890, −3.145383490602325, −2.698880476727741, −1.416418044236686, −0.5526345736666898,
0.5526345736666898, 1.416418044236686, 2.698880476727741, 3.145383490602325, 3.983665932993890, 5.062487669256276, 5.344497141337892, 6.487682379506930, 6.990154917484951, 7.747167180777181, 8.180617064320379, 8.547787014779639, 9.606504628214356, 9.907243586677620, 10.78804382853390, 11.03739139641071, 11.89453166528835, 12.17835000536460, 13.01194329839553, 13.48716178919939, 14.27197356621564, 15.04738475060442, 15.33754606729125, 16.04263550250978, 16.28993897294991
