| L(s) = 1 | − 1.61·2-s + 0.618·3-s + 0.618·4-s − 0.618·5-s − 1.00·6-s − 7-s + 2.23·8-s − 2.61·9-s + 1.00·10-s − 11-s + 0.381·12-s − 1.76·13-s + 1.61·14-s − 0.381·15-s − 4.85·16-s + 0.236·17-s + 4.23·18-s − 3.85·19-s − 0.381·20-s − 0.618·21-s + 1.61·22-s − 1.38·23-s + 1.38·24-s − 4.61·25-s + 2.85·26-s − 3.47·27-s − 0.618·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.356·3-s + 0.309·4-s − 0.276·5-s − 0.408·6-s − 0.377·7-s + 0.790·8-s − 0.872·9-s + 0.316·10-s − 0.301·11-s + 0.110·12-s − 0.489·13-s + 0.432·14-s − 0.0986·15-s − 1.21·16-s + 0.0572·17-s + 0.998·18-s − 0.884·19-s − 0.0854·20-s − 0.134·21-s + 0.344·22-s − 0.288·23-s + 0.282·24-s − 0.923·25-s + 0.559·26-s − 0.668·27-s − 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 19 | \( 1 + 3.85T + 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 - 0.854T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 - 4.61T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 0.527T + 61T^{2} \) |
| 67 | \( 1 + 0.909T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 7.94T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 - 3.47T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−11.08267367468802642398819168410, −10.19800128145452940706623265426, −9.359695630173813234106596060774, −8.489232094052008884558852424876, −7.85168181858551961766210150419, −6.73666697600104912432448076844, −5.29805219343833053366075221404, −3.81217979207927878359753821501, −2.23799494825827285624372484241, 0,
2.23799494825827285624372484241, 3.81217979207927878359753821501, 5.29805219343833053366075221404, 6.73666697600104912432448076844, 7.85168181858551961766210150419, 8.489232094052008884558852424876, 9.359695630173813234106596060774, 10.19800128145452940706623265426, 11.08267367468802642398819168410
