| L(s) = 1 | + 5-s − 3·7-s − 2·11-s − 4·13-s − 5·17-s − 5·19-s − 4·23-s − 4·25-s − 29-s + 8·31-s − 3·35-s + 3·37-s − 9·41-s + 5·43-s − 3·47-s + 2·49-s + 10·53-s − 2·55-s + 7·59-s − 10·61-s − 4·65-s − 8·67-s + 4·71-s − 14·73-s + 6·77-s − 14·79-s + 8·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 0.603·11-s − 1.10·13-s − 1.21·17-s − 1.14·19-s − 0.834·23-s − 4/5·25-s − 0.185·29-s + 1.43·31-s − 0.507·35-s + 0.493·37-s − 1.40·41-s + 0.762·43-s − 0.437·47-s + 2/7·49-s + 1.37·53-s − 0.269·55-s + 0.911·59-s − 1.28·61-s − 0.496·65-s − 0.977·67-s + 0.474·71-s − 1.63·73-s + 0.683·77-s − 1.57·79-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4443790827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4443790827\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−15.88490598168728, −15.32583221518762, −14.98640548349818, −14.16511981762862, −13.53896306918969, −13.17446743194421, −12.73726106260303, −11.98009684864908, −11.61763201364625, −10.57640050797170, −10.26948353394747, −9.757251547288293, −9.206267085092682, −8.501217181950894, −7.922298058758054, −7.095351417100848, −6.595890086426275, −6.042426086314950, −5.430127105308361, −4.484902095504705, −4.114183245028064, −2.977691043578631, −2.490074002615354, −1.805233334863304, −0.2622564430122089,
0.2622564430122089, 1.805233334863304, 2.490074002615354, 2.977691043578631, 4.114183245028064, 4.484902095504705, 5.430127105308361, 6.042426086314950, 6.595890086426275, 7.095351417100848, 7.922298058758054, 8.501217181950894, 9.206267085092682, 9.757251547288293, 10.26948353394747, 10.57640050797170, 11.61763201364625, 11.98009684864908, 12.73726106260303, 13.17446743194421, 13.53896306918969, 14.16511981762862, 14.98640548349818, 15.32583221518762, 15.88490598168728
