| L(s) = 1 | − 2-s − 4-s + 3·8-s + 11-s − 2·13-s − 16-s − 2·17-s + 6·19-s − 22-s + 4·23-s + 2·26-s + 6·29-s − 4·31-s − 5·32-s + 2·34-s + 6·37-s − 6·38-s − 10·41-s − 6·43-s − 44-s − 4·46-s + 8·47-s + 2·52-s − 6·58-s + 4·59-s + 6·61-s + 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 0.301·11-s − 0.554·13-s − 1/4·16-s − 0.485·17-s + 1.37·19-s − 0.213·22-s + 0.834·23-s + 0.392·26-s + 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.973·38-s − 1.56·41-s − 0.914·43-s − 0.150·44-s − 0.589·46-s + 1.16·47-s + 0.277·52-s − 0.787·58-s + 0.520·59-s + 0.768·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.76422728785076, −13.34842248713922, −12.89801873191256, −12.36216250434860, −11.67803545791018, −11.46941312142821, −10.76924563686559, −10.20398213738419, −9.883812724158129, −9.430028171734906, −8.823422093998279, −8.589178171874851, −7.964405634277958, −7.299504217449136, −7.110118802518183, −6.461043486132126, −5.609482996926929, −5.234924329165480, −4.619166026249695, −4.225570484855435, −3.409450389268805, −2.931522960622610, −2.109681355481439, −1.360286151157253, −0.8286405881870288, 0,
0.8286405881870288, 1.360286151157253, 2.109681355481439, 2.931522960622610, 3.409450389268805, 4.225570484855435, 4.619166026249695, 5.234924329165480, 5.609482996926929, 6.461043486132126, 7.110118802518183, 7.299504217449136, 7.964405634277958, 8.589178171874851, 8.823422093998279, 9.430028171734906, 9.883812724158129, 10.20398213738419, 10.76924563686559, 11.46941312142821, 11.67803545791018, 12.36216250434860, 12.89801873191256, 13.34842248713922, 13.76422728785076
