| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s − 2·11-s − 6·13-s − 15-s + 2·19-s + 2·21-s + 8·23-s + 25-s − 27-s + 2·33-s − 2·35-s + 2·37-s + 6·39-s + 41-s − 4·43-s + 45-s + 2·47-s − 3·49-s + 6·53-s − 2·55-s − 2·57-s − 12·59-s − 2·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.258·15-s + 0.458·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.348·33-s − 0.338·35-s + 0.328·37-s + 0.960·39-s + 0.156·41-s − 0.609·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.824·53-s − 0.269·55-s − 0.264·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 41 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−15.12168483411465, −14.65244127524048, −13.92574622787675, −13.42290104597047, −12.91933275222396, −12.45343635270586, −12.11283085675179, −11.32533306973837, −10.89214826752685, −10.19660853684487, −9.904959373401144, −9.295402095773182, −8.940611799860508, −7.867542950595062, −7.547348792286664, −6.759467556481006, −6.573573813922435, −5.649476546299966, −5.158612381366410, −4.831379697490899, −3.973507218841824, −2.983968420658068, −2.731274399389520, −1.813057192818321, −0.8299127827594105, 0,
0.8299127827594105, 1.813057192818321, 2.731274399389520, 2.983968420658068, 3.973507218841824, 4.831379697490899, 5.158612381366410, 5.649476546299966, 6.573573813922435, 6.759467556481006, 7.547348792286664, 7.867542950595062, 8.940611799860508, 9.295402095773182, 9.904959373401144, 10.19660853684487, 10.89214826752685, 11.32533306973837, 12.11283085675179, 12.45343635270586, 12.91933275222396, 13.42290104597047, 13.92574622787675, 14.65244127524048, 15.12168483411465
