| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s − 5·13-s + 15-s − 5·17-s + 19-s − 3·21-s − 4·23-s + 25-s − 27-s − 5·29-s + 2·31-s − 3·35-s − 3·37-s + 5·39-s − 2·41-s + 2·43-s − 45-s + 6·47-s + 2·49-s + 5·51-s − 57-s − 4·59-s + 8·61-s + 3·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.38·13-s + 0.258·15-s − 1.21·17-s + 0.229·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 0.359·31-s − 0.507·35-s − 0.493·37-s + 0.800·39-s − 0.312·41-s + 0.304·43-s − 0.149·45-s + 0.875·47-s + 2/7·49-s + 0.700·51-s − 0.132·57-s − 0.520·59-s + 1.02·61-s + 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7905586262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7905586262\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.66272816184232, −13.08395333642099, −12.39016569954091, −12.12368430758164, −11.71304941905169, −11.13634601843291, −10.84717328021619, −10.34223141565895, −9.603617332051079, −9.341461769468741, −8.537939011995914, −8.134484344638824, −7.601959644429112, −7.169211760014138, −6.684115703897888, −6.028631820623362, −5.271841122388252, −5.074757926936345, −4.363667411223507, −4.099502918340353, −3.265066602308833, −2.295724639095432, −2.076987600702136, −1.194861632124857, −0.2933671159111409,
0.2933671159111409, 1.194861632124857, 2.076987600702136, 2.295724639095432, 3.265066602308833, 4.099502918340353, 4.363667411223507, 5.074757926936345, 5.271841122388252, 6.028631820623362, 6.684115703897888, 7.169211760014138, 7.601959644429112, 8.134484344638824, 8.537939011995914, 9.341461769468741, 9.603617332051079, 10.34223141565895, 10.84717328021619, 11.13634601843291, 11.71304941905169, 12.12368430758164, 12.39016569954091, 13.08395333642099, 13.66272816184232
