| L(s) = 1 | + 2-s − 2.32·3-s + 4-s − 3.20·5-s − 2.32·6-s + 3.53·7-s + 8-s + 2.39·9-s − 3.20·10-s − 4.03·11-s − 2.32·12-s − 0.696·13-s + 3.53·14-s + 7.44·15-s + 16-s − 0.319·17-s + 2.39·18-s + 0.520·19-s − 3.20·20-s − 8.20·21-s − 4.03·22-s + 2.84·23-s − 2.32·24-s + 5.29·25-s − 0.696·26-s + 1.41·27-s + 3.53·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·3-s + 0.5·4-s − 1.43·5-s − 0.947·6-s + 1.33·7-s + 0.353·8-s + 0.797·9-s − 1.01·10-s − 1.21·11-s − 0.670·12-s − 0.193·13-s + 0.944·14-s + 1.92·15-s + 0.250·16-s − 0.0775·17-s + 0.563·18-s + 0.119·19-s − 0.717·20-s − 1.79·21-s − 0.860·22-s + 0.593·23-s − 0.473·24-s + 1.05·25-s − 0.136·26-s + 0.271·27-s + 0.667·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.32T + 3T^{2} \) |
| 5 | \( 1 + 3.20T + 5T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 + 0.696T + 13T^{2} \) |
| 17 | \( 1 + 0.319T + 17T^{2} \) |
| 19 | \( 1 - 0.520T + 19T^{2} \) |
| 23 | \( 1 - 2.84T + 23T^{2} \) |
| 29 | \( 1 - 2.77T + 29T^{2} \) |
| 31 | \( 1 - 0.557T + 31T^{2} \) |
| 37 | \( 1 + 8.21T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 + 7.49T + 43T^{2} \) |
| 47 | \( 1 - 0.915T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 8.36T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 7.94T + 67T^{2} \) |
| 71 | \( 1 + 2.97T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 - 6.75T + 79T^{2} \) |
| 83 | \( 1 + 3.49T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 8.29T + 97T^{2} \) |
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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
−7.50643329396456331708812045395, −7.12407087084852067388089579805, −6.11812142759536824166032823817, −5.30406122082633796999765971005, −4.88560832135309490590064028987, −4.40925164778611255278853451945, −3.44609195282672842493108801269, −2.42872518050204455033321550126, −1.12064512645171193409384328378, 0,
1.12064512645171193409384328378, 2.42872518050204455033321550126, 3.44609195282672842493108801269, 4.40925164778611255278853451945, 4.88560832135309490590064028987, 5.30406122082633796999765971005, 6.11812142759536824166032823817, 7.12407087084852067388089579805, 7.50643329396456331708812045395
