| L(s) = 1 | − 3-s + 7-s + 9-s + 6·11-s − 4·13-s − 6·17-s + 2·19-s − 21-s − 27-s − 6·29-s + 10·31-s − 6·33-s + 2·37-s + 4·39-s − 6·41-s + 4·43-s + 49-s + 6·51-s − 12·53-s − 2·57-s − 14·61-s + 63-s + 4·67-s − 6·71-s + 4·73-s + 6·77-s + 16·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 1.45·17-s + 0.458·19-s − 0.218·21-s − 0.192·27-s − 1.11·29-s + 1.79·31-s − 1.04·33-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.840·51-s − 1.64·53-s − 0.264·57-s − 1.79·61-s + 0.125·63-s + 0.488·67-s − 0.712·71-s + 0.468·73-s + 0.683·77-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−15.13414256644215, −14.88238893894425, −14.16191579271802, −13.74610056642836, −13.20539693200989, −12.38655170949019, −12.05009508587890, −11.62302205683398, −11.10368365760069, −10.61774346397741, −9.814061268173627, −9.336441140020254, −9.029367896281939, −8.174752766869331, −7.580299892892183, −6.968576089087269, −6.369867972773065, −6.136472276457090, −5.025644782397572, −4.730911880731072, −4.119404709118497, −3.425889717262776, −2.462097622319795, −1.763773410984981, −1.020037317932297, 0,
1.020037317932297, 1.763773410984981, 2.462097622319795, 3.425889717262776, 4.119404709118497, 4.730911880731072, 5.025644782397572, 6.136472276457090, 6.369867972773065, 6.968576089087269, 7.580299892892183, 8.174752766869331, 9.029367896281939, 9.336441140020254, 9.814061268173627, 10.61774346397741, 11.10368365760069, 11.62302205683398, 12.05009508587890, 12.38655170949019, 13.20539693200989, 13.74610056642836, 14.16191579271802, 14.88238893894425, 15.13414256644215
