| L(s) = 1 | + 5-s − 4·7-s + 4·11-s − 3·17-s − 4·23-s − 4·25-s + 29-s − 4·31-s − 4·35-s + 3·37-s + 9·41-s + 8·43-s − 8·47-s + 9·49-s + 9·53-s + 4·55-s − 4·59-s + 7·61-s − 4·67-s − 8·71-s + 11·73-s − 16·77-s + 4·79-s − 3·85-s + 6·89-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.20·11-s − 0.727·17-s − 0.834·23-s − 4/5·25-s + 0.185·29-s − 0.718·31-s − 0.676·35-s + 0.493·37-s + 1.40·41-s + 1.21·43-s − 1.16·47-s + 9/7·49-s + 1.23·53-s + 0.539·55-s − 0.520·59-s + 0.896·61-s − 0.488·67-s − 0.949·71-s + 1.28·73-s − 1.82·77-s + 0.450·79-s − 0.325·85-s + 0.635·89-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.86554337143794, −15.12233862579585, −14.54465878041012, −14.04416383675603, −13.43310573876203, −13.08617542946017, −12.42650821265589, −12.02798347832963, −11.32303721532986, −10.77381549198470, −10.01216320097203, −9.601446584455177, −9.219753291533630, −8.693356253563824, −7.825794904244357, −7.144015445998991, −6.589812952378070, −6.031734995341245, −5.786882239433483, −4.669420729273332, −3.932576398791135, −3.591989417146627, −2.619270803176160, −2.044078689033782, −1.001413622533431, 0,
1.001413622533431, 2.044078689033782, 2.619270803176160, 3.591989417146627, 3.932576398791135, 4.669420729273332, 5.786882239433483, 6.031734995341245, 6.589812952378070, 7.144015445998991, 7.825794904244357, 8.693356253563824, 9.219753291533630, 9.601446584455177, 10.01216320097203, 10.77381549198470, 11.32303721532986, 12.02798347832963, 12.42650821265589, 13.08617542946017, 13.43310573876203, 14.04416383675603, 14.54465878041012, 15.12233862579585, 15.86554337143794
