| L(s) = 1 | + 2-s + 0.790·3-s + 4-s + 1.22·5-s + 0.790·6-s − 2.48·7-s + 8-s − 2.37·9-s + 1.22·10-s + 0.0747·11-s + 0.790·12-s + 0.162·13-s − 2.48·14-s + 0.970·15-s + 16-s + 2.22·17-s − 2.37·18-s − 19-s + 1.22·20-s − 1.96·21-s + 0.0747·22-s + 1.30·23-s + 0.790·24-s − 3.49·25-s + 0.162·26-s − 4.24·27-s − 2.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.456·3-s + 0.5·4-s + 0.548·5-s + 0.322·6-s − 0.939·7-s + 0.353·8-s − 0.791·9-s + 0.388·10-s + 0.0225·11-s + 0.228·12-s + 0.0451·13-s − 0.664·14-s + 0.250·15-s + 0.250·16-s + 0.538·17-s − 0.559·18-s − 0.229·19-s + 0.274·20-s − 0.429·21-s + 0.0159·22-s + 0.272·23-s + 0.161·24-s − 0.698·25-s + 0.0319·26-s − 0.817·27-s − 0.469·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.790T + 3T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 + 2.48T + 7T^{2} \) |
| 11 | \( 1 - 0.0747T + 11T^{2} \) |
| 13 | \( 1 - 0.162T + 13T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 + 5.51T + 31T^{2} \) |
| 37 | \( 1 - 12.0T + 37T^{2} \) |
| 41 | \( 1 - 3.75T + 41T^{2} \) |
| 43 | \( 1 - 2.46T + 43T^{2} \) |
| 47 | \( 1 + 8.33T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 1.89T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 5.28T + 73T^{2} \) |
| 79 | \( 1 - 0.0962T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 - 9.61T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.65060868367902143336018112974, −6.49992909829990007667063727120, −6.05282061785666352457075087772, −5.56714366209693026529512720109, −4.64990606384075998357304969781, −3.70527331477194115754340406245, −3.16597346797152533377652428599, −2.47028163411981421300439894274, −1.57760137227454764808345633751, 0,
1.57760137227454764808345633751, 2.47028163411981421300439894274, 3.16597346797152533377652428599, 3.70527331477194115754340406245, 4.64990606384075998357304969781, 5.56714366209693026529512720109, 6.05282061785666352457075087772, 6.49992909829990007667063727120, 7.65060868367902143336018112974
