| L(s) = 1 | − 1.69·2-s + 3-s + 0.875·4-s − 0.871·5-s − 1.69·6-s + 1.90·8-s + 9-s + 1.47·10-s − 0.879·11-s + 0.875·12-s − 6.61·13-s − 0.871·15-s − 4.98·16-s − 5.48·17-s − 1.69·18-s + 3.14·19-s − 0.762·20-s + 1.49·22-s − 8.57·23-s + 1.90·24-s − 4.24·25-s + 11.2·26-s + 27-s + 0.898·29-s + 1.47·30-s − 9.92·31-s + 4.63·32-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.577·3-s + 0.437·4-s − 0.389·5-s − 0.692·6-s + 0.674·8-s + 0.333·9-s + 0.467·10-s − 0.265·11-s + 0.252·12-s − 1.83·13-s − 0.224·15-s − 1.24·16-s − 1.33·17-s − 0.399·18-s + 0.721·19-s − 0.170·20-s + 0.317·22-s − 1.78·23-s + 0.389·24-s − 0.848·25-s + 2.19·26-s + 0.192·27-s + 0.166·29-s + 0.269·30-s − 1.78·31-s + 0.819·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4416682200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4416682200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 5 | \( 1 + 0.871T + 5T^{2} \) |
| 11 | \( 1 + 0.879T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 + 8.57T + 23T^{2} \) |
| 29 | \( 1 - 0.898T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 43 | \( 1 - 1.22T + 43T^{2} \) |
| 47 | \( 1 + 4.45T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 + 6.59T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 + 8.87T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 0.181T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 6.65T + 89T^{2} \) |
| 97 | \( 1 - 2.66T + 97T^{2} \) |
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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
−7.87307244264909006672277146786, −7.71662299064612182380759513239, −7.14274946608600226514656920097, −6.16606553935995476548603779387, −5.04486206681802409877034126246, −4.41562597443163958820095866332, −3.61520046472643106212852661850, −2.32565503584429645333259466034, −1.96297413144694461362702429108, −0.38571943161191844602192312708,
0.38571943161191844602192312708, 1.96297413144694461362702429108, 2.32565503584429645333259466034, 3.61520046472643106212852661850, 4.41562597443163958820095866332, 5.04486206681802409877034126246, 6.16606553935995476548603779387, 7.14274946608600226514656920097, 7.71662299064612182380759513239, 7.87307244264909006672277146786
