| L(s) = 1 | + 2-s + 1.13·3-s + 4-s + 1.28·5-s + 1.13·6-s − 4.07·7-s + 8-s − 1.70·9-s + 1.28·10-s + 1.29·11-s + 1.13·12-s + 4.65·13-s − 4.07·14-s + 1.46·15-s + 16-s − 6.04·17-s − 1.70·18-s + 0.245·19-s + 1.28·20-s − 4.63·21-s + 1.29·22-s − 5.67·23-s + 1.13·24-s − 3.33·25-s + 4.65·26-s − 5.35·27-s − 4.07·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.656·3-s + 0.5·4-s + 0.576·5-s + 0.464·6-s − 1.54·7-s + 0.353·8-s − 0.568·9-s + 0.407·10-s + 0.389·11-s + 0.328·12-s + 1.29·13-s − 1.08·14-s + 0.378·15-s + 0.250·16-s − 1.46·17-s − 0.402·18-s + 0.0563·19-s + 0.288·20-s − 1.01·21-s + 0.275·22-s − 1.18·23-s + 0.232·24-s − 0.667·25-s + 0.913·26-s − 1.03·27-s − 0.770·28-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 | \( 1 - 1.13T + 3T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 + 6.04T + 17T^{2} \) |
| 19 | \( 1 - 0.245T + 19T^{2} \) |
| 23 | \( 1 + 5.67T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 + 5.59T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 + 6.75T + 53T^{2} \) |
| 59 | \( 1 - 0.975T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 8.19T + 71T^{2} \) |
| 73 | \( 1 - 5.43T + 73T^{2} \) |
| 79 | \( 1 - 8.00T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 4.98T + 89T^{2} \) |
| 97 | \( 1 + 1.85T + 97T^{2} \) |
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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
−7.75468618718798941621081109801, −6.55653157196106519004891601845, −6.34831341599968124194494949259, −5.85171162193308454198650219831, −4.74682162063262118749023990634, −3.68951884786474242929064361333, −3.44283268243114969464517210219, −2.48057805624700811129656241204, −1.73746306056196791828593645850, 0,
1.73746306056196791828593645850, 2.48057805624700811129656241204, 3.44283268243114969464517210219, 3.68951884786474242929064361333, 4.74682162063262118749023990634, 5.85171162193308454198650219831, 6.34831341599968124194494949259, 6.55653157196106519004891601845, 7.75468618718798941621081109801
