| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·10-s − 2·11-s − 4·16-s + 19-s − 2·20-s − 4·22-s − 3·23-s − 4·25-s + 5·29-s + 9·31-s − 8·32-s + 2·38-s − 2·41-s − 43-s − 4·44-s − 6·46-s − 3·47-s − 8·50-s + 9·53-s + 2·55-s + 10·58-s + 2·61-s + 18·62-s − 8·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.632·10-s − 0.603·11-s − 16-s + 0.229·19-s − 0.447·20-s − 0.852·22-s − 0.625·23-s − 4/5·25-s + 0.928·29-s + 1.61·31-s − 1.41·32-s + 0.324·38-s − 0.312·41-s − 0.152·43-s − 0.603·44-s − 0.884·46-s − 0.437·47-s − 1.13·50-s + 1.23·53-s + 0.269·55-s + 1.31·58-s + 0.256·61-s + 2.28·62-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 17 T + p T^{2} \) | 1.89.ar |
| 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
−14.16940022725485, −13.67569689279427, −13.50866085297756, −12.92563388596723, −12.28153052839424, −11.90597011905642, −11.69148434370937, −11.01052534976701, −10.30039800315468, −10.03318026840079, −9.249339856673781, −8.644589782165219, −8.069108301389665, −7.644100532010595, −6.942926425563668, −6.346591180774055, −5.985239448002013, −5.292918641339022, −4.794919321382419, −4.360409525888555, −3.715956871114792, −3.229236285202131, −2.566671165788765, −2.047285907757490, −0.9302119741686822, 0,
0.9302119741686822, 2.047285907757490, 2.566671165788765, 3.229236285202131, 3.715956871114792, 4.360409525888555, 4.794919321382419, 5.292918641339022, 5.985239448002013, 6.346591180774055, 6.942926425563668, 7.644100532010595, 8.069108301389665, 8.644589782165219, 9.249339856673781, 10.03318026840079, 10.30039800315468, 11.01052534976701, 11.69148434370937, 11.90597011905642, 12.28153052839424, 12.92563388596723, 13.50866085297756, 13.67569689279427, 14.16940022725485
