| L(s) = 1 | + 5-s − 4·11-s − 5·13-s + 5·17-s − 8·19-s − 4·23-s − 4·25-s − 3·29-s + 4·31-s + 3·37-s + 6·41-s − 4·43-s − 12·47-s − 7·49-s + 10·53-s − 4·55-s + 8·59-s − 5·61-s − 5·65-s − 8·67-s + 16·71-s − 5·73-s − 4·79-s + 4·83-s + 5·85-s − 3·89-s − 8·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 1.38·13-s + 1.21·17-s − 1.83·19-s − 0.834·23-s − 4/5·25-s − 0.557·29-s + 0.718·31-s + 0.493·37-s + 0.937·41-s − 0.609·43-s − 1.75·47-s − 49-s + 1.37·53-s − 0.539·55-s + 1.04·59-s − 0.640·61-s − 0.620·65-s − 0.977·67-s + 1.89·71-s − 0.585·73-s − 0.450·79-s + 0.439·83-s + 0.542·85-s − 0.317·89-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−9.503898964438682367280900598625, −8.170842358458906967841960826649, −7.85105010987388363519424464059, −6.75155461247528669892324601661, −5.82199868237918594961477276173, −5.09539883020764153687479254698, −4.12402291918165081446882489944, −2.78062191844038234402813668067, −1.97554871893638655702914851568, 0,
1.97554871893638655702914851568, 2.78062191844038234402813668067, 4.12402291918165081446882489944, 5.09539883020764153687479254698, 5.82199868237918594961477276173, 6.75155461247528669892324601661, 7.85105010987388363519424464059, 8.170842358458906967841960826649, 9.503898964438682367280900598625
