| L(s) = 1 | − 1.06·2-s − 0.856·4-s + 0.420·5-s + 1.13·7-s + 3.05·8-s − 0.449·10-s − 2.91·11-s − 1.70·13-s − 1.21·14-s − 1.55·16-s + 0.536·17-s + 0.585·19-s − 0.360·20-s + 3.11·22-s + 23-s − 4.82·25-s + 1.81·26-s − 0.971·28-s + 29-s + 7.49·31-s − 4.44·32-s − 0.573·34-s + 0.477·35-s − 1.22·37-s − 0.626·38-s + 1.28·40-s + 12.2·41-s + ⋯ |
| L(s) = 1 | − 0.756·2-s − 0.428·4-s + 0.188·5-s + 0.428·7-s + 1.07·8-s − 0.142·10-s − 0.879·11-s − 0.471·13-s − 0.324·14-s − 0.388·16-s + 0.130·17-s + 0.134·19-s − 0.0806·20-s + 0.665·22-s + 0.208·23-s − 0.964·25-s + 0.356·26-s − 0.183·28-s + 0.185·29-s + 1.34·31-s − 0.786·32-s − 0.0983·34-s + 0.0806·35-s − 0.201·37-s − 0.101·38-s + 0.203·40-s + 1.90·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.06T + 2T^{2} \) |
| 5 | \( 1 - 0.420T + 5T^{2} \) |
| 7 | \( 1 - 1.13T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 0.536T + 17T^{2} \) |
| 19 | \( 1 - 0.585T + 19T^{2} \) |
| 31 | \( 1 - 7.49T + 31T^{2} \) |
| 37 | \( 1 + 1.22T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 - 0.180T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 5.86T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 4.56T + 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 + 8.57T + 83T^{2} \) |
| 89 | \( 1 - 2.75T + 89T^{2} \) |
| 97 | \( 1 + 7.28T + 97T^{2} \) |
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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
−7.83233160858933192400508973322, −7.38755680781498496961335715671, −6.35042150471493109258493014908, −5.50464822609535095829701775238, −4.80666982074736455554649703314, −4.24666658446354029378790121115, −3.07799688548930854532430095106, −2.15226690947954985385620552200, −1.15258697518118553789476517391, 0,
1.15258697518118553789476517391, 2.15226690947954985385620552200, 3.07799688548930854532430095106, 4.24666658446354029378790121115, 4.80666982074736455554649703314, 5.50464822609535095829701775238, 6.35042150471493109258493014908, 7.38755680781498496961335715671, 7.83233160858933192400508973322
