| L(s) = 1 | − 2·3-s + 5-s + 4·7-s + 9-s + 2·11-s + 13-s − 2·15-s + 2·17-s − 6·19-s − 8·21-s + 6·23-s + 25-s + 4·27-s − 2·29-s + 10·31-s − 4·33-s + 4·35-s + 2·37-s − 2·39-s − 6·41-s + 10·43-s + 45-s − 4·47-s + 9·49-s − 4·51-s − 2·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.516·15-s + 0.485·17-s − 1.37·19-s − 1.74·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s + 1.79·31-s − 0.696·33-s + 0.676·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 1.52·43-s + 0.149·45-s − 0.583·47-s + 9/7·49-s − 0.560·51-s − 0.274·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.797839104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797839104\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−8.486165620646531108447977597785, −7.66952683561360827989011694472, −6.72055609364729716086042023517, −6.17408739217622025418478327959, −5.43098161002914470593834212398, −4.77443775775279551995461388430, −4.20526671591691528901862730924, −2.81620198757012189665961257056, −1.67937651122023764644502455937, −0.874535190542795796910703920322,
0.874535190542795796910703920322, 1.67937651122023764644502455937, 2.81620198757012189665961257056, 4.20526671591691528901862730924, 4.77443775775279551995461388430, 5.43098161002914470593834212398, 6.17408739217622025418478327959, 6.72055609364729716086042023517, 7.66952683561360827989011694472, 8.486165620646531108447977597785
