| L(s) = 1 | + 3-s + 9-s + 2·11-s + 13-s + 4·17-s − 4·19-s − 4·23-s + 27-s − 8·31-s + 2·33-s + 2·37-s + 39-s + 10·41-s + 4·43-s − 6·47-s − 7·49-s + 4·51-s − 6·53-s − 4·57-s + 14·59-s − 10·61-s − 4·67-s − 4·69-s + 8·71-s + 8·73-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s + 0.192·27-s − 1.43·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s + 1.56·41-s + 0.609·43-s − 0.875·47-s − 49-s + 0.560·51-s − 0.824·53-s − 0.529·57-s + 1.82·59-s − 1.28·61-s − 0.488·67-s − 0.481·69-s + 0.949·71-s + 0.936·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.90038775589189, −13.21765937980840, −12.79213373803427, −12.47058488228847, −11.88553763721686, −11.31794694908273, −10.85186642810764, −10.40600542743860, −9.672550287165395, −9.458941930920868, −8.922627384943832, −8.328582735211383, −7.865296562866127, −7.523945470996350, −6.778338540621514, −6.318213017122941, −5.817837607839294, −5.218782154106537, −4.508499757137145, −3.928000690807664, −3.609523427022375, −2.897367024653551, −2.215957495048094, −1.650628811682512, −0.9835572104669278, 0,
0.9835572104669278, 1.650628811682512, 2.215957495048094, 2.897367024653551, 3.609523427022375, 3.928000690807664, 4.508499757137145, 5.218782154106537, 5.817837607839294, 6.318213017122941, 6.778338540621514, 7.523945470996350, 7.865296562866127, 8.328582735211383, 8.922627384943832, 9.458941930920868, 9.672550287165395, 10.40600542743860, 10.85186642810764, 11.31794694908273, 11.88553763721686, 12.47058488228847, 12.79213373803427, 13.21765937980840, 13.90038775589189
