| L(s) = 1 | − 5-s − 2·7-s + 4·11-s + 4·13-s − 17-s + 4·19-s + 4·23-s + 25-s + 10·29-s + 2·35-s + 2·37-s − 4·41-s + 10·43-s − 3·49-s + 2·53-s − 4·55-s + 2·59-s + 2·61-s − 4·65-s − 6·67-s − 6·71-s − 8·73-s − 8·77-s − 4·79-s + 16·83-s + 85-s − 14·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s + 1.10·13-s − 0.242·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.338·35-s + 0.328·37-s − 0.624·41-s + 1.52·43-s − 3/7·49-s + 0.274·53-s − 0.539·55-s + 0.260·59-s + 0.256·61-s − 0.496·65-s − 0.733·67-s − 0.712·71-s − 0.936·73-s − 0.911·77-s − 0.450·79-s + 1.75·83-s + 0.108·85-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.259433599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.259433599\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−16.19466072749426, −15.94728714517394, −15.28342638488729, −14.68344146702868, −13.98168847901613, −13.60698603049447, −12.92421454223704, −12.30122056536326, −11.78062751563521, −11.25749714580760, −10.63388200485970, −9.936540128150880, −9.285819894507699, −8.796998747599434, −8.250111979031079, −7.359086475003327, −6.769300478708924, −6.293938841780331, −5.618422868484421, −4.634719289462718, −4.044408319175151, −3.315571507862891, −2.781644427578691, −1.417995784035817, −0.7600514988023767,
0.7600514988023767, 1.417995784035817, 2.781644427578691, 3.315571507862891, 4.044408319175151, 4.634719289462718, 5.618422868484421, 6.293938841780331, 6.769300478708924, 7.359086475003327, 8.250111979031079, 8.796998747599434, 9.285819894507699, 9.936540128150880, 10.63388200485970, 11.25749714580760, 11.78062751563521, 12.30122056536326, 12.92421454223704, 13.60698603049447, 13.98168847901613, 14.68344146702868, 15.28342638488729, 15.94728714517394, 16.19466072749426
