L(s) = 1 | − 0.679·2-s − 3-s − 1.53·4-s + 2.83·5-s + 0.679·6-s − 0.718·7-s + 2.40·8-s + 9-s − 1.92·10-s − 0.429·11-s + 1.53·12-s + 0.378·13-s + 0.488·14-s − 2.83·15-s + 1.44·16-s + 17-s − 0.679·18-s − 0.638·19-s − 4.35·20-s + 0.718·21-s + 0.291·22-s + 4.51·23-s − 2.40·24-s + 3.01·25-s − 0.257·26-s − 27-s + 1.10·28-s + ⋯ |
L(s) = 1 | − 0.480·2-s − 0.577·3-s − 0.768·4-s + 1.26·5-s + 0.277·6-s − 0.271·7-s + 0.850·8-s + 0.333·9-s − 0.608·10-s − 0.129·11-s + 0.443·12-s + 0.105·13-s + 0.130·14-s − 0.730·15-s + 0.360·16-s + 0.242·17-s − 0.160·18-s − 0.146·19-s − 0.973·20-s + 0.156·21-s + 0.0622·22-s + 0.942·23-s − 0.490·24-s + 0.602·25-s − 0.0505·26-s − 0.192·27-s + 0.208·28-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 | \( 1 + 0.679T + 2T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 + 0.718T + 7T^{2} \) |
| 11 | \( 1 + 0.429T + 11T^{2} \) |
| 13 | \( 1 - 0.378T + 13T^{2} \) |
| 19 | \( 1 + 0.638T + 19T^{2} \) |
| 23 | \( 1 - 4.51T + 23T^{2} \) |
| 29 | \( 1 - 0.186T + 29T^{2} \) |
| 31 | \( 1 + 6.25T + 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 + 3.44T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 + 7.10T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 - 1.34T + 61T^{2} \) |
| 67 | \( 1 + 0.865T + 67T^{2} \) |
| 71 | \( 1 - 8.28T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 8.22T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 4.45T + 97T^{2} \) |
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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
−7.52096000031611252937930819182, −6.71381562593564881850865183070, −6.08620490210244481634688069290, −5.31478638355218653184183500317, −4.97060401460764871075871870894, −3.98381800729208326427345626941, −3.07740077459449010290436214282, −1.91540379519828004765634187019, −1.18730710758410228851540384567, 0,
1.18730710758410228851540384567, 1.91540379519828004765634187019, 3.07740077459449010290436214282, 3.98381800729208326427345626941, 4.97060401460764871075871870894, 5.31478638355218653184183500317, 6.08620490210244481634688069290, 6.71381562593564881850865183070, 7.52096000031611252937930819182