| L(s) = 1 | + 2·3-s − 4·5-s + 9-s − 4·11-s + 4·13-s − 8·15-s + 6·17-s + 2·19-s − 23-s + 11·25-s − 4·27-s + 6·29-s + 8·31-s − 8·33-s − 10·37-s + 8·39-s − 6·41-s − 12·43-s − 4·45-s + 8·47-s + 12·51-s + 2·53-s + 16·55-s + 4·57-s − 10·59-s − 4·61-s − 16·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 1/3·9-s − 1.20·11-s + 1.10·13-s − 2.06·15-s + 1.45·17-s + 0.458·19-s − 0.208·23-s + 11/5·25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s − 1.39·33-s − 1.64·37-s + 1.28·39-s − 0.937·41-s − 1.82·43-s − 0.596·45-s + 1.16·47-s + 1.68·51-s + 0.274·53-s + 2.15·55-s + 0.529·57-s − 1.30·59-s − 0.512·61-s − 1.98·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.19188115479191, −13.87540353419684, −13.63661694060096, −12.78144528254462, −12.34423678912405, −11.88852668144793, −11.47101588755548, −10.83650583960227, −10.22065402456415, −9.979717064684435, −9.027868529833747, −8.414771457917784, −8.291083137006230, −7.925782674353985, −7.362832234902389, −6.854360282401172, −6.073525653038296, −5.252478074992635, −4.854808004706705, −4.010065766106710, −3.569573918257881, −3.059448538940605, −2.813170917956076, −1.670910159741078, −0.8908703996823039, 0,
0.8908703996823039, 1.670910159741078, 2.813170917956076, 3.059448538940605, 3.569573918257881, 4.010065766106710, 4.854808004706705, 5.252478074992635, 6.073525653038296, 6.854360282401172, 7.362832234902389, 7.925782674353985, 8.291083137006230, 8.414771457917784, 9.027868529833747, 9.979717064684435, 10.22065402456415, 10.83650583960227, 11.47101588755548, 11.88852668144793, 12.34423678912405, 12.78144528254462, 13.63661694060096, 13.87540353419684, 14.19188115479191
