| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s − 2·9-s − 5·11-s + 12-s − 4·13-s − 16-s + 5·17-s + 2·18-s + 7·19-s + 5·22-s + 9·23-s − 3·24-s − 5·25-s + 4·26-s + 5·27-s − 6·29-s − 5·32-s + 5·33-s − 5·34-s + 2·36-s + 2·37-s − 7·38-s + 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s − 2/3·9-s − 1.50·11-s + 0.288·12-s − 1.10·13-s − 1/4·16-s + 1.21·17-s + 0.471·18-s + 1.60·19-s + 1.06·22-s + 1.87·23-s − 0.612·24-s − 25-s + 0.784·26-s + 0.962·27-s − 1.11·29-s − 0.883·32-s + 0.870·33-s − 0.857·34-s + 1/3·36-s + 0.328·37-s − 1.13·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2989 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2989 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) | |
| 61 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.276308518186042246301121116411, −7.57225698586501730555949075144, −7.32467701430304040457171363886, −5.80221863560049773222755509516, −5.25105947275816693832766178702, −4.83834375315262096804429262427, −3.41234100838815627421664551588, −2.56873908396028409297284428013, −1.04409936087548574536101154677, 0,
1.04409936087548574536101154677, 2.56873908396028409297284428013, 3.41234100838815627421664551588, 4.83834375315262096804429262427, 5.25105947275816693832766178702, 5.80221863560049773222755509516, 7.32467701430304040457171363886, 7.57225698586501730555949075144, 8.276308518186042246301121116411
