| L(s) = 1 | + 2·3-s + 9-s − 2·17-s + 2·19-s + 8·23-s − 4·27-s − 2·29-s + 4·31-s − 6·37-s + 2·41-s − 8·43-s + 4·47-s − 4·51-s − 10·53-s + 4·57-s − 6·59-s + 4·61-s + 12·67-s + 16·69-s − 14·73-s + 8·79-s − 11·81-s + 6·83-s − 4·87-s − 10·89-s + 8·93-s − 2·97-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.485·17-s + 0.458·19-s + 1.66·23-s − 0.769·27-s − 0.371·29-s + 0.718·31-s − 0.986·37-s + 0.312·41-s − 1.21·43-s + 0.583·47-s − 0.560·51-s − 1.37·53-s + 0.529·57-s − 0.781·59-s + 0.512·61-s + 1.46·67-s + 1.92·69-s − 1.63·73-s + 0.900·79-s − 1.22·81-s + 0.658·83-s − 0.428·87-s − 1.05·89-s + 0.829·93-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−14.24578431051427, −13.80083776033785, −13.36882128979001, −12.91061280133318, −12.43574488859028, −11.70115021842303, −11.28874235894922, −10.80026102660086, −10.11854546878722, −9.649628766281337, −9.082615867709727, −8.748565477788141, −8.305896490924472, −7.613275472213121, −7.293607264554542, −6.597999909081538, −6.122522706345531, −5.170254146539751, −4.964119406264824, −4.094115589461707, −3.493516790296033, −3.021741390164949, −2.492893953540129, −1.777251469366038, −1.071907909150218, 0,
1.071907909150218, 1.777251469366038, 2.492893953540129, 3.021741390164949, 3.493516790296033, 4.094115589461707, 4.964119406264824, 5.170254146539751, 6.122522706345531, 6.597999909081538, 7.293607264554542, 7.613275472213121, 8.305896490924472, 8.748565477788141, 9.082615867709727, 9.649628766281337, 10.11854546878722, 10.80026102660086, 11.28874235894922, 11.70115021842303, 12.43574488859028, 12.91061280133318, 13.36882128979001, 13.80083776033785, 14.24578431051427
