| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 12-s + 2·13-s + 14-s + 16-s + 4·17-s + 18-s − 6·19-s − 21-s − 4·23-s − 24-s − 5·25-s + 2·26-s − 27-s + 28-s + 10·29-s + 6·31-s + 32-s + 4·34-s + 36-s − 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.37·19-s − 0.218·21-s − 0.834·23-s − 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.85·29-s + 1.07·31-s + 0.176·32-s + 0.685·34-s + 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.804826958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.804826958\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.051813753089857686428291149334, −7.50195258614813863797879007663, −6.42489977862829018225919109306, −6.10098520284764516138950755290, −5.36900763128811539114773250289, −4.40989342728991428553771509734, −4.05576474933775594859751576814, −2.90878683357439788919878297904, −1.96324335591924275791437827569, −0.866846976089747955444418497930,
0.866846976089747955444418497930, 1.96324335591924275791437827569, 2.90878683357439788919878297904, 4.05576474933775594859751576814, 4.40989342728991428553771509734, 5.36900763128811539114773250289, 6.10098520284764516138950755290, 6.42489977862829018225919109306, 7.50195258614813863797879007663, 8.051813753089857686428291149334
