| L(s) = 1 | − 1.45·3-s + 2.84·5-s − 2.57·7-s − 0.880·9-s + 1.01·11-s − 2.90·13-s − 4.14·15-s − 0.407·17-s − 19-s + 3.74·21-s + 0.705·23-s + 3.10·25-s + 5.64·27-s + 3.60·29-s + 7.00·31-s − 1.47·33-s − 7.32·35-s − 7.51·37-s + 4.22·39-s + 3.63·41-s + 8.45·43-s − 2.50·45-s + 3.72·47-s − 0.381·49-s + 0.593·51-s + 3.14·53-s + 2.87·55-s + ⋯ |
| L(s) = 1 | − 0.840·3-s + 1.27·5-s − 0.972·7-s − 0.293·9-s + 0.304·11-s − 0.805·13-s − 1.07·15-s − 0.0989·17-s − 0.229·19-s + 0.817·21-s + 0.147·23-s + 0.621·25-s + 1.08·27-s + 0.669·29-s + 1.25·31-s − 0.255·33-s − 1.23·35-s − 1.23·37-s + 0.676·39-s + 0.567·41-s + 1.28·43-s − 0.373·45-s + 0.543·47-s − 0.0544·49-s + 0.0831·51-s + 0.432·53-s + 0.387·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 1.45T + 3T^{2} \) |
| 5 | \( 1 - 2.84T + 5T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 + 0.407T + 17T^{2} \) |
| 23 | \( 1 - 0.705T + 23T^{2} \) |
| 29 | \( 1 - 3.60T + 29T^{2} \) |
| 31 | \( 1 - 7.00T + 31T^{2} \) |
| 37 | \( 1 + 7.51T + 37T^{2} \) |
| 41 | \( 1 - 3.63T + 41T^{2} \) |
| 43 | \( 1 - 8.45T + 43T^{2} \) |
| 47 | \( 1 - 3.72T + 47T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 1.16T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 6.99T + 73T^{2} \) |
| 83 | \( 1 - 4.86T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 + 7.50T + 97T^{2} \) |
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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
−7.53629966119752078287382304836, −6.69109326356841387881826029001, −6.26839543874515266378687719048, −5.71756521147019515096836894207, −5.04727408508234924607861231440, −4.21858266896373554957435879640, −2.97098222730018772406434882709, −2.44029336858613503453611047387, −1.20763001413457132477081616216, 0,
1.20763001413457132477081616216, 2.44029336858613503453611047387, 2.97098222730018772406434882709, 4.21858266896373554957435879640, 5.04727408508234924607861231440, 5.71756521147019515096836894207, 6.26839543874515266378687719048, 6.69109326356841387881826029001, 7.53629966119752078287382304836
