| L(s) = 1 | − 3-s + 4·5-s − 2·7-s + 9-s + 4·11-s − 2·13-s − 4·15-s − 2·17-s + 8·19-s + 2·21-s − 4·23-s + 11·25-s − 27-s + 6·31-s − 4·33-s − 8·35-s + 2·37-s + 2·39-s + 6·41-s + 4·45-s − 4·47-s − 3·49-s + 2·51-s + 16·55-s − 8·57-s − 4·59-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 1.03·15-s − 0.485·17-s + 1.83·19-s + 0.436·21-s − 0.834·23-s + 11/5·25-s − 0.192·27-s + 1.07·31-s − 0.696·33-s − 1.35·35-s + 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.596·45-s − 0.583·47-s − 3/7·49-s + 0.280·51-s + 2.15·55-s − 1.05·57-s − 0.520·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.483899732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.483899732\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−11.38766792869140056495258951368, −10.06632064709018516311308621993, −9.738751180865856613138829349946, −8.985911500488081519481984584802, −7.27371486188202795087403560747, −6.30170361535159036673173480205, −5.81778647559970487461786585347, −4.60632905768716122167488407077, −2.92708427914109016867182117304, −1.43950309180346771366044691508,
1.43950309180346771366044691508, 2.92708427914109016867182117304, 4.60632905768716122167488407077, 5.81778647559970487461786585347, 6.30170361535159036673173480205, 7.27371486188202795087403560747, 8.985911500488081519481984584802, 9.738751180865856613138829349946, 10.06632064709018516311308621993, 11.38766792869140056495258951368
