| L(s) = 1 | + 2·3-s + 9-s + 6·17-s − 2·19-s − 5·25-s − 4·27-s − 6·41-s + 10·43-s − 7·49-s + 12·51-s − 4·57-s + 6·59-s − 14·67-s + 2·73-s − 10·75-s − 11·81-s − 18·83-s − 18·89-s + 10·97-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.45·17-s − 0.458·19-s − 25-s − 0.769·27-s − 0.937·41-s + 1.52·43-s − 49-s + 1.68·51-s − 0.529·57-s + 0.781·59-s − 1.71·67-s + 0.234·73-s − 1.15·75-s − 1.22·81-s − 1.97·83-s − 1.90·89-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.22840110151183, −14.78581210420147, −14.20273883279464, −13.99299222651327, −13.35332845942104, −12.77215470918862, −12.31538597238103, −11.59362390843264, −11.21496398013580, −10.26056352193107, −9.991416366307068, −9.388569810320400, −8.786332083027887, −8.342672317517355, −7.704060608193627, −7.440677359741891, −6.577959901334849, −5.820060670759920, −5.429750158535054, −4.462978696998440, −3.878673723116833, −3.256573953076998, −2.724351351180553, −1.956563988817826, −1.242816694801125, 0,
1.242816694801125, 1.956563988817826, 2.724351351180553, 3.256573953076998, 3.878673723116833, 4.462978696998440, 5.429750158535054, 5.820060670759920, 6.577959901334849, 7.440677359741891, 7.704060608193627, 8.342672317517355, 8.786332083027887, 9.388569810320400, 9.991416366307068, 10.26056352193107, 11.21496398013580, 11.59362390843264, 12.31538597238103, 12.77215470918862, 13.35332845942104, 13.99299222651327, 14.20273883279464, 14.78581210420147, 15.22840110151183
