| 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.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 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.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 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.g |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 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
−13.74028830665728, −13.38155683879010, −13.16282389172483, −12.44827513334585, −11.96276414773392, −11.31519057046292, −11.02474215422190, −9.955844172096328, −9.882816481527718, −9.560339166316355, −8.903825671707965, −8.461460205413326, −7.910814062001428, −7.505340013244575, −6.943612850014560, −6.259567023424140, −5.812042732784820, −5.365896204412937, −4.423780642338376, −4.068918898196106, −3.269723620374374, −2.952414272092372, −2.332098979723100, −1.851170625814323, −0.9252417931596189, 0,
0.9252417931596189, 1.851170625814323, 2.332098979723100, 2.952414272092372, 3.269723620374374, 4.068918898196106, 4.423780642338376, 5.365896204412937, 5.812042732784820, 6.259567023424140, 6.943612850014560, 7.505340013244575, 7.910814062001428, 8.461460205413326, 8.903825671707965, 9.560339166316355, 9.882816481527718, 9.955844172096328, 11.02474215422190, 11.31519057046292, 11.96276414773392, 12.44827513334585, 13.16282389172483, 13.38155683879010, 13.74028830665728
