| L(s) = 1 | − 2·3-s + 5-s + 2·7-s + 9-s + 11-s − 2·15-s − 3·17-s − 6·19-s − 4·21-s + 4·23-s − 4·25-s + 4·27-s − 29-s − 4·31-s − 2·33-s + 2·35-s + 9·37-s + 41-s − 4·43-s + 45-s + 6·47-s − 3·49-s + 6·51-s + 9·53-s + 55-s + 12·57-s − 6·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.516·15-s − 0.727·17-s − 1.37·19-s − 0.872·21-s + 0.834·23-s − 4/5·25-s + 0.769·27-s − 0.185·29-s − 0.718·31-s − 0.348·33-s + 0.338·35-s + 1.47·37-s + 0.156·41-s − 0.609·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.840·51-s + 1.23·53-s + 0.134·55-s + 1.58·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.85624878879279, −13.25492599713768, −12.64488572384478, −12.52018300972561, −11.64406618788922, −11.39555902031628, −10.97513021825376, −10.64308786097752, −10.03741064778618, −9.430136751704463, −8.937705927377619, −8.443337672994811, −7.897984998140709, −7.245614916698947, −6.652582025507069, −6.301493600022591, −5.730652705349025, −5.344444070791340, −4.674369576169565, −4.327574997263542, −3.690973486325801, −2.715496332837208, −2.157952644018660, −1.547176711443001, −0.7776132580910384, 0,
0.7776132580910384, 1.547176711443001, 2.157952644018660, 2.715496332837208, 3.690973486325801, 4.327574997263542, 4.674369576169565, 5.344444070791340, 5.730652705349025, 6.301493600022591, 6.652582025507069, 7.245614916698947, 7.897984998140709, 8.443337672994811, 8.937705927377619, 9.430136751704463, 10.03741064778618, 10.64308786097752, 10.97513021825376, 11.39555902031628, 11.64406618788922, 12.52018300972561, 12.64488572384478, 13.25492599713768, 13.85624878879279
