| L(s) = 1 | − 4·5-s − 4·11-s − 6·13-s − 6·17-s + 6·19-s − 8·23-s + 11·25-s + 6·29-s − 4·31-s + 10·37-s − 2·41-s + 10·43-s − 4·47-s − 7·49-s − 12·53-s + 16·55-s + 4·59-s − 2·61-s + 24·65-s − 10·67-s + 8·71-s − 2·73-s − 10·79-s − 83-s + 24·85-s + 8·89-s − 24·95-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.20·11-s − 1.66·13-s − 1.45·17-s + 1.37·19-s − 1.66·23-s + 11/5·25-s + 1.11·29-s − 0.718·31-s + 1.64·37-s − 0.312·41-s + 1.52·43-s − 0.583·47-s − 49-s − 1.64·53-s + 2.15·55-s + 0.520·59-s − 0.256·61-s + 2.97·65-s − 1.22·67-s + 0.949·71-s − 0.234·73-s − 1.12·79-s − 0.109·83-s + 2.60·85-s + 0.847·89-s − 2.46·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.84172789071974, −14.44344277338093, −13.91070727732164, −13.09259008249417, −12.74688480082572, −12.14836576482397, −11.79429451264103, −11.31192681741961, −10.79301459163678, −10.22331946254964, −9.582768320397396, −9.139868913631717, −8.205554626276628, −7.866301303356573, −7.652222885841392, −7.005759671028429, −6.420315447479326, −5.515201955576252, −4.906821303870147, −4.417658971797660, −4.039529995150327, −2.925109111636876, −2.831006540706792, −1.863114792173401, −0.5630582753442937, 0,
0.5630582753442937, 1.863114792173401, 2.831006540706792, 2.925109111636876, 4.039529995150327, 4.417658971797660, 4.906821303870147, 5.515201955576252, 6.420315447479326, 7.005759671028429, 7.652222885841392, 7.866301303356573, 8.205554626276628, 9.139868913631717, 9.582768320397396, 10.22331946254964, 10.79301459163678, 11.31192681741961, 11.79429451264103, 12.14836576482397, 12.74688480082572, 13.09259008249417, 13.91070727732164, 14.44344277338093, 14.84172789071974
