| L(s) = 1 | + 3-s − 5-s − 7-s − 2·9-s + 2·13-s − 15-s + 2·19-s − 21-s + 25-s − 5·27-s − 6·29-s − 4·31-s + 35-s − 4·37-s + 2·39-s + 9·41-s − 43-s + 2·45-s − 3·47-s − 6·49-s − 6·53-s + 2·57-s − 61-s + 2·63-s − 2·65-s − 13·67-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.554·13-s − 0.258·15-s + 0.458·19-s − 0.218·21-s + 1/5·25-s − 0.962·27-s − 1.11·29-s − 0.718·31-s + 0.169·35-s − 0.657·37-s + 0.320·39-s + 1.40·41-s − 0.152·43-s + 0.298·45-s − 0.437·47-s − 6/7·49-s − 0.824·53-s + 0.264·57-s − 0.128·61-s + 0.251·63-s − 0.248·65-s − 1.58·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.681970147137822419741882729066, −7.78667265407983692284995200613, −7.26352560215082484752070882399, −6.16320187908263068669150930956, −5.54554295076159684748913929965, −4.40673448495562041266774317554, −3.48512686951098685014029052902, −2.91525132795625078984260251203, −1.64813837338244602720297241843, 0,
1.64813837338244602720297241843, 2.91525132795625078984260251203, 3.48512686951098685014029052902, 4.40673448495562041266774317554, 5.54554295076159684748913929965, 6.16320187908263068669150930956, 7.26352560215082484752070882399, 7.78667265407983692284995200613, 8.681970147137822419741882729066
