| L(s) = 1 | − 3-s − 2·9-s + 3·13-s − 2·17-s − 8·19-s + 4·23-s − 5·25-s + 5·27-s + 3·29-s + 2·31-s − 6·37-s − 3·39-s + 8·43-s + 12·47-s + 2·51-s − 10·53-s + 8·57-s − 9·59-s + 7·61-s − 9·67-s − 4·69-s + 14·71-s − 14·73-s + 5·75-s − 9·79-s + 81-s − 2·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.832·13-s − 0.485·17-s − 1.83·19-s + 0.834·23-s − 25-s + 0.962·27-s + 0.557·29-s + 0.359·31-s − 0.986·37-s − 0.480·39-s + 1.21·43-s + 1.75·47-s + 0.280·51-s − 1.37·53-s + 1.05·57-s − 1.17·59-s + 0.896·61-s − 1.09·67-s − 0.481·69-s + 1.66·71-s − 1.63·73-s + 0.577·75-s − 1.01·79-s + 1/9·81-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9543366515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9543366515\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−14.51655687650268, −14.03264801100090, −13.62492155109751, −12.93151535859100, −12.54393910371661, −11.99804825974365, −11.40003515652982, −10.90677887278290, −10.66178551084331, −10.06341335531179, −9.188349856661079, −8.766299156914326, −8.420699385701311, −7.723176376229593, −6.986489788902729, −6.442809425424686, −5.963689788934877, −5.584122489678334, −4.665746963386891, −4.316045297891922, −3.556012388121465, −2.790880011620695, −2.168121009419204, −1.302565338775655, −0.3708820094512695,
0.3708820094512695, 1.302565338775655, 2.168121009419204, 2.790880011620695, 3.556012388121465, 4.316045297891922, 4.665746963386891, 5.584122489678334, 5.963689788934877, 6.442809425424686, 6.986489788902729, 7.723176376229593, 8.420699385701311, 8.766299156914326, 9.188349856661079, 10.06341335531179, 10.66178551084331, 10.90677887278290, 11.40003515652982, 11.99804825974365, 12.54393910371661, 12.93151535859100, 13.62492155109751, 14.03264801100090, 14.51655687650268
