| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 5·11-s − 15-s − 2·17-s − 5·19-s + 21-s + 2·23-s + 25-s − 27-s − 6·29-s + 3·31-s + 5·33-s − 35-s − 8·37-s + 7·41-s − 4·43-s + 45-s − 3·47-s + 49-s + 2·51-s − 2·53-s − 5·55-s + 5·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.258·15-s − 0.485·17-s − 1.14·19-s + 0.218·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.538·31-s + 0.870·33-s − 0.169·35-s − 1.31·37-s + 1.09·41-s − 0.609·43-s + 0.149·45-s − 0.437·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 0.674·55-s + 0.662·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−14.29491124383798, −13.77417081036495, −13.21985254238291, −12.87779967841532, −12.55496743736525, −11.93778815498255, −11.10634110082847, −10.92369863890178, −10.39549736613820, −9.957573041115233, −9.357597076092120, −8.808865567613538, −8.254882315767221, −7.597094990242243, −7.180168896515351, −6.354131849913048, −6.195242891780028, −5.431902501977854, −4.941714968863638, −4.524454256169951, −3.620333523365509, −3.064103385101260, −2.200465437408138, −1.898519699401510, −0.6993085577924110, 0,
0.6993085577924110, 1.898519699401510, 2.200465437408138, 3.064103385101260, 3.620333523365509, 4.524454256169951, 4.941714968863638, 5.431902501977854, 6.195242891780028, 6.354131849913048, 7.180168896515351, 7.597094990242243, 8.254882315767221, 8.808865567613538, 9.357597076092120, 9.957573041115233, 10.39549736613820, 10.92369863890178, 11.10634110082847, 11.93778815498255, 12.55496743736525, 12.87779967841532, 13.21985254238291, 13.77417081036495, 14.29491124383798
