| L(s) = 1 | − 2-s − 0.618·3-s + 4-s + 5-s + 0.618·6-s − 7-s − 8-s − 2.61·9-s − 10-s − 0.618·12-s − 3.38·13-s + 14-s − 0.618·15-s + 16-s − 2.38·17-s + 2.61·18-s + 0.763·19-s + 20-s + 0.618·21-s − 0.472·23-s + 0.618·24-s + 25-s + 3.38·26-s + 3.47·27-s − 28-s − 0.854·29-s + 0.618·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.356·3-s + 0.5·4-s + 0.447·5-s + 0.252·6-s − 0.377·7-s − 0.353·8-s − 0.872·9-s − 0.316·10-s − 0.178·12-s − 0.937·13-s + 0.267·14-s − 0.159·15-s + 0.250·16-s − 0.577·17-s + 0.617·18-s + 0.175·19-s + 0.223·20-s + 0.134·21-s − 0.0984·23-s + 0.126·24-s + 0.200·25-s + 0.663·26-s + 0.668·27-s − 0.188·28-s − 0.158·29-s + 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 13 | \( 1 + 3.38T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 + 0.854T + 29T^{2} \) |
| 31 | \( 1 - 0.763T + 31T^{2} \) |
| 37 | \( 1 - 9.70T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 + 2.47T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 - 3.23T + 67T^{2} \) |
| 71 | \( 1 - 8.32T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 5.85T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 5.61T + 97T^{2} \) |
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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
−7.51435016862333867799071201758, −6.70328894985533100531360673576, −6.18252509838450941638277428647, −5.52063715643670033074036937191, −4.80472356323487439056386726753, −3.81224564744162149699986728843, −2.67033290476247438678675169710, −2.36865826761148818421962458724, −1.01410183985042093900036901219, 0,
1.01410183985042093900036901219, 2.36865826761148818421962458724, 2.67033290476247438678675169710, 3.81224564744162149699986728843, 4.80472356323487439056386726753, 5.52063715643670033074036937191, 6.18252509838450941638277428647, 6.70328894985533100531360673576, 7.51435016862333867799071201758
