| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 15-s − 2·19-s + 2·21-s + 6·23-s + 25-s + 27-s − 4·31-s − 2·35-s + 2·37-s + 6·41-s − 4·43-s − 45-s − 3·49-s + 6·53-s − 2·57-s + 10·61-s + 2·63-s − 8·67-s + 6·69-s − 8·73-s + 75-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.258·15-s − 0.458·19-s + 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.718·31-s − 0.338·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 3/7·49-s + 0.824·53-s − 0.264·57-s + 1.28·61-s + 0.251·63-s − 0.977·67-s + 0.722·69-s − 0.936·73-s + 0.115·75-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.40623587373347, −12.99866096009113, −12.75160517618853, −11.98489205331024, −11.59864776199220, −11.16925581949428, −10.63407979318350, −10.29318408303371, −9.549655632215293, −9.083529237294824, −8.712899622724399, −8.178775773100913, −7.760068312601398, −7.277664996232322, −6.803705226592192, −6.244627896317679, −5.416633447285365, −5.126447488389406, −4.362617545979461, −4.089252441538601, −3.374188219315211, −2.808561203232312, −2.251657105991709, −1.521782749239839, −0.9702981374938472, 0,
0.9702981374938472, 1.521782749239839, 2.251657105991709, 2.808561203232312, 3.374188219315211, 4.089252441538601, 4.362617545979461, 5.126447488389406, 5.416633447285365, 6.244627896317679, 6.803705226592192, 7.277664996232322, 7.760068312601398, 8.178775773100913, 8.712899622724399, 9.083529237294824, 9.549655632215293, 10.29318408303371, 10.63407979318350, 11.16925581949428, 11.59864776199220, 11.98489205331024, 12.75160517618853, 12.99866096009113, 13.40623587373347
