| L(s) = 1 | − 3-s + 7-s + 9-s − 2·13-s − 19-s − 21-s + 6·23-s − 27-s + 31-s − 8·37-s + 2·39-s − 3·41-s − 8·43-s − 6·49-s + 12·53-s + 57-s − 9·59-s + 8·61-s + 63-s + 4·67-s − 6·69-s − 9·71-s + 4·73-s − 10·79-s + 81-s + 6·83-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.229·19-s − 0.218·21-s + 1.25·23-s − 0.192·27-s + 0.179·31-s − 1.31·37-s + 0.320·39-s − 0.468·41-s − 1.21·43-s − 6/7·49-s + 1.64·53-s + 0.132·57-s − 1.17·59-s + 1.02·61-s + 0.125·63-s + 0.488·67-s − 0.722·69-s − 1.06·71-s + 0.468·73-s − 1.12·79-s + 1/9·81-s + 0.658·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−7.19355470650795061213466103967, −6.79517447019648239316074806642, −5.99404537863425312207554445380, −5.08842342225329171779444287378, −4.89697736325595639702402626063, −3.88700895097995562143120085316, −3.06697229521996234079893871551, −2.08428063699289600908954654082, −1.18723507337409373225495669628, 0,
1.18723507337409373225495669628, 2.08428063699289600908954654082, 3.06697229521996234079893871551, 3.88700895097995562143120085316, 4.89697736325595639702402626063, 5.08842342225329171779444287378, 5.99404537863425312207554445380, 6.79517447019648239316074806642, 7.19355470650795061213466103967
