| L(s) = 1 | − 2-s + 4-s + 4·5-s + 4·7-s − 8-s − 4·10-s − 4·11-s + 5·13-s − 4·14-s + 16-s − 4·17-s + 3·19-s + 4·20-s + 4·22-s + 23-s + 11·25-s − 5·26-s + 4·28-s + 3·29-s − 11·31-s − 32-s + 4·34-s + 16·35-s + 2·37-s − 3·38-s − 4·40-s − 12·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 1.51·7-s − 0.353·8-s − 1.26·10-s − 1.20·11-s + 1.38·13-s − 1.06·14-s + 1/4·16-s − 0.970·17-s + 0.688·19-s + 0.894·20-s + 0.852·22-s + 0.208·23-s + 11/5·25-s − 0.980·26-s + 0.755·28-s + 0.557·29-s − 1.97·31-s − 0.176·32-s + 0.685·34-s + 2.70·35-s + 0.328·37-s − 0.486·38-s − 0.632·40-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.15283585859653, −12.72515867178349, −11.85004415796051, −11.49139183852593, −10.88236609248935, −10.77525960126918, −10.27890494156113, −9.822503002458216, −9.202203420265166, −8.878130066949305, −8.381777544031266, −8.088780381723213, −7.454386734688155, −6.764355538731018, −6.596931024698590, −5.658790298654915, −5.486381353099560, −5.180672966909690, −4.471349495679646, −3.762888241904569, −2.937191466874082, −2.473195474435289, −1.893434367782746, −1.500271439815015, −1.075610775317073, 0,
1.075610775317073, 1.500271439815015, 1.893434367782746, 2.473195474435289, 2.937191466874082, 3.762888241904569, 4.471349495679646, 5.180672966909690, 5.486381353099560, 5.658790298654915, 6.596931024698590, 6.764355538731018, 7.454386734688155, 8.088780381723213, 8.381777544031266, 8.878130066949305, 9.202203420265166, 9.822503002458216, 10.27890494156113, 10.77525960126918, 10.88236609248935, 11.49139183852593, 11.85004415796051, 12.72515867178349, 13.15283585859653
