| L(s) = 1 | − 2-s + 4-s − 8-s − 13-s + 16-s + 6·17-s + 4·19-s − 6·23-s − 5·25-s + 26-s − 8·31-s − 32-s − 6·34-s + 2·37-s − 4·38-s − 4·43-s + 6·46-s + 6·47-s + 5·50-s − 52-s + 6·59-s − 2·61-s + 8·62-s + 64-s − 4·67-s + 6·68-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 1.25·23-s − 25-s + 0.196·26-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.648·38-s − 0.609·43-s + 0.884·46-s + 0.875·47-s + 0.707·50-s − 0.138·52-s + 0.781·59-s − 0.256·61-s + 1.01·62-s + 1/8·64-s − 0.488·67-s + 0.727·68-s + 1.17·73-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
| \(L(1)\) |
\(\approx\) |
\(1.297666390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.297666390\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−16.44553907172931, −16.09604213370029, −15.37764107327769, −14.80939726214759, −14.17792888476827, −13.74496448258758, −12.92482491937725, −12.15777770738053, −11.92912993290686, −11.22695902053414, −10.50405568878316, −9.908570703432551, −9.563331301793379, −8.880291079783424, −7.998051805762789, −7.691205635144370, −7.125433338926979, −6.173539959309209, −5.653919403613001, −5.017160700124632, −3.856543293576988, −3.393783985721077, −2.356843149538121, −1.610547786899652, −0.5918100691217246,
0.5918100691217246, 1.610547786899652, 2.356843149538121, 3.393783985721077, 3.856543293576988, 5.017160700124632, 5.653919403613001, 6.173539959309209, 7.125433338926979, 7.691205635144370, 7.998051805762789, 8.880291079783424, 9.563331301793379, 9.908570703432551, 10.50405568878316, 11.22695902053414, 11.92912993290686, 12.15777770738053, 12.92482491937725, 13.74496448258758, 14.17792888476827, 14.80939726214759, 15.37764107327769, 16.09604213370029, 16.44553907172931
