| L(s) = 1 | − 2-s − 0.618·3-s + 4-s + 0.618·5-s + 0.618·6-s − 3.85·7-s − 8-s − 2.61·9-s − 0.618·10-s − 0.618·12-s + 2.23·13-s + 3.85·14-s − 0.381·15-s + 16-s + 17-s + 2.61·18-s + 6.47·19-s + 0.618·20-s + 2.38·21-s − 6.23·23-s + 0.618·24-s − 4.61·25-s − 2.23·26-s + 3.47·27-s − 3.85·28-s + 9.23·29-s + 0.381·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.356·3-s + 0.5·4-s + 0.276·5-s + 0.252·6-s − 1.45·7-s − 0.353·8-s − 0.872·9-s − 0.195·10-s − 0.178·12-s + 0.620·13-s + 1.03·14-s − 0.0986·15-s + 0.250·16-s + 0.242·17-s + 0.617·18-s + 1.48·19-s + 0.138·20-s + 0.519·21-s − 1.30·23-s + 0.126·24-s − 0.923·25-s − 0.438·26-s + 0.668·27-s − 0.728·28-s + 1.71·29-s + 0.0697·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 9.23T + 29T^{2} \) |
| 31 | \( 1 + 7.47T + 31T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 - 9.70T + 41T^{2} \) |
| 43 | \( 1 - 4.38T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 + 7.76T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 7.61T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 1.09T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 5.47T + 97T^{2} \) |
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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
−7.976166847944267568214406742069, −7.49052839071738258839428892069, −6.33826600967817400099697773315, −6.09553074970172506961500323542, −5.45443619516312674638572694824, −4.08247301930276280380083935220, −3.17473940223919000387504003680, −2.54072658222304017118301829953, −1.11629425467493395567316341408, 0,
1.11629425467493395567316341408, 2.54072658222304017118301829953, 3.17473940223919000387504003680, 4.08247301930276280380083935220, 5.45443619516312674638572694824, 6.09553074970172506961500323542, 6.33826600967817400099697773315, 7.49052839071738258839428892069, 7.976166847944267568214406742069
