| L(s) = 1 | − 2·5-s + 7-s + 11-s + 2·13-s − 6·17-s + 8·19-s + 4·23-s − 25-s − 2·29-s − 8·31-s − 2·35-s + 6·37-s − 6·41-s − 8·43-s + 4·47-s + 49-s − 10·53-s − 2·55-s + 4·59-s − 14·61-s − 4·65-s + 4·67-s − 4·71-s − 14·73-s + 77-s + 8·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.301·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 1.37·53-s − 0.269·55-s + 0.520·59-s − 1.79·61-s − 0.496·65-s + 0.488·67-s − 0.474·71-s − 1.63·73-s + 0.113·77-s + 0.900·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−16.74073726960675, −15.96226300461547, −15.81194145102517, −15.02389446993866, −14.69754359574697, −13.83456003713443, −13.37099910849743, −12.85283430402046, −11.93157671460129, −11.57766896369497, −11.14342599509863, −10.57456541541796, −9.634642119626678, −9.052529265383199, −8.616951931820084, −7.643352133727088, −7.483031256436544, −6.633596769424717, −5.927254849785841, −5.036504508249632, −4.554759410967863, −3.629621983993562, −3.237177386965924, −2.048957245382733, −1.170604523763546, 0,
1.170604523763546, 2.048957245382733, 3.237177386965924, 3.629621983993562, 4.554759410967863, 5.036504508249632, 5.927254849785841, 6.633596769424717, 7.483031256436544, 7.643352133727088, 8.616951931820084, 9.052529265383199, 9.634642119626678, 10.57456541541796, 11.14342599509863, 11.57766896369497, 11.93157671460129, 12.85283430402046, 13.37099910849743, 13.83456003713443, 14.69754359574697, 15.02389446993866, 15.81194145102517, 15.96226300461547, 16.74073726960675
