L(s) = 1 | + 1.51·2-s + 3-s + 0.283·4-s − 0.836·5-s + 1.51·6-s + 4.41·7-s − 2.59·8-s + 9-s − 1.26·10-s − 3.77·11-s + 0.283·12-s − 5.83·13-s + 6.66·14-s − 0.836·15-s − 4.48·16-s − 17-s + 1.51·18-s + 4.09·19-s − 0.236·20-s + 4.41·21-s − 5.70·22-s + 9.55·23-s − 2.59·24-s − 4.30·25-s − 8.82·26-s + 27-s + 1.24·28-s + ⋯ |
L(s) = 1 | + 1.06·2-s + 0.577·3-s + 0.141·4-s − 0.373·5-s + 0.616·6-s + 1.66·7-s − 0.917·8-s + 0.333·9-s − 0.399·10-s − 1.13·11-s + 0.0816·12-s − 1.61·13-s + 1.78·14-s − 0.215·15-s − 1.12·16-s − 0.242·17-s + 0.356·18-s + 0.939·19-s − 0.0529·20-s + 0.962·21-s − 1.21·22-s + 1.99·23-s − 0.529·24-s − 0.860·25-s − 1.73·26-s + 0.192·27-s + 0.235·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 - 1.51T + 2T^{2} \) |
| 5 | \( 1 + 0.836T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 + 5.83T + 13T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 - 9.55T + 23T^{2} \) |
| 29 | \( 1 + 9.47T + 29T^{2} \) |
| 31 | \( 1 - 0.516T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 - 3.76T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 1.99T + 53T^{2} \) |
| 59 | \( 1 - 3.16T + 59T^{2} \) |
| 61 | \( 1 + 6.64T + 61T^{2} \) |
| 67 | \( 1 + 2.31T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 + 7.20T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.49061832216513408491753083009, −7.04395290698845611221240852358, −5.61343564385124068973938978879, −5.18830741019234897558429953184, −4.73851827034418491457647713921, −4.10224779237025297305352251913, −3.07628632073417436689626157044, −2.55857199675575979606460891621, −1.58131104531063412740141268616, 0,
1.58131104531063412740141268616, 2.55857199675575979606460891621, 3.07628632073417436689626157044, 4.10224779237025297305352251913, 4.73851827034418491457647713921, 5.18830741019234897558429953184, 5.61343564385124068973938978879, 7.04395290698845611221240852358, 7.49061832216513408491753083009