| L(s) = 1 | + 3-s + 9-s − 11-s + 2·13-s − 4·17-s − 2·19-s − 5·25-s + 27-s − 4·29-s + 2·31-s − 33-s − 37-s + 2·39-s + 10·41-s + 2·43-s + 8·47-s − 7·49-s − 4·51-s + 10·53-s − 2·57-s − 10·61-s + 8·67-s − 16·71-s + 10·73-s − 5·75-s + 14·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.970·17-s − 0.458·19-s − 25-s + 0.192·27-s − 0.742·29-s + 0.359·31-s − 0.174·33-s − 0.164·37-s + 0.320·39-s + 1.56·41-s + 0.304·43-s + 1.16·47-s − 49-s − 0.560·51-s + 1.37·53-s − 0.264·57-s − 1.28·61-s + 0.977·67-s − 1.89·71-s + 1.17·73-s − 0.577·75-s + 1.57·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.90444127421554, −15.28757496789667, −15.06027254614426, −14.26881866920561, −13.66716931848766, −13.42620804270563, −12.71832012324761, −12.27182437328149, −11.45058429404321, −10.95206752034526, −10.49536264025876, −9.700895416884941, −9.226836219477109, −8.672659803994774, −8.087782328763402, −7.524529319196411, −6.912238515462627, −6.146776425561176, −5.681517956263061, −4.750658085067436, −4.103195178190372, −3.622039823920595, −2.608542847144417, −2.160221202839836, −1.190152853545647, 0,
1.190152853545647, 2.160221202839836, 2.608542847144417, 3.622039823920595, 4.103195178190372, 4.750658085067436, 5.681517956263061, 6.146776425561176, 6.912238515462627, 7.524529319196411, 8.087782328763402, 8.672659803994774, 9.226836219477109, 9.700895416884941, 10.49536264025876, 10.95206752034526, 11.45058429404321, 12.27182437328149, 12.71832012324761, 13.42620804270563, 13.66716931848766, 14.26881866920561, 15.06027254614426, 15.28757496789667, 15.90444127421554
