L(s) = 1 | − 1.80·2-s + 3-s + 1.24·4-s + 5-s − 1.80·6-s − 7-s + 1.35·8-s + 9-s − 1.80·10-s + 1.80·11-s + 1.24·12-s + 1.80·14-s + 15-s − 4.93·16-s − 2.44·17-s − 1.80·18-s + 5.74·19-s + 1.24·20-s − 21-s − 3.24·22-s − 7.04·23-s + 1.35·24-s − 4·25-s + 27-s − 1.24·28-s − 3.15·29-s − 1.80·30-s + ⋯ |
L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.623·4-s + 0.447·5-s − 0.735·6-s − 0.377·7-s + 0.479·8-s + 0.333·9-s − 0.569·10-s + 0.543·11-s + 0.359·12-s + 0.481·14-s + 0.258·15-s − 1.23·16-s − 0.593·17-s − 0.424·18-s + 1.31·19-s + 0.278·20-s − 0.218·21-s − 0.692·22-s − 1.46·23-s + 0.276·24-s − 0.800·25-s + 0.192·27-s − 0.235·28-s − 0.586·29-s − 0.328·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 + 7.04T + 23T^{2} \) |
| 29 | \( 1 + 3.15T + 29T^{2} \) |
| 31 | \( 1 + 0.445T + 31T^{2} \) |
| 37 | \( 1 + 3.98T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 - 8.92T + 53T^{2} \) |
| 59 | \( 1 + 9.71T + 59T^{2} \) |
| 61 | \( 1 + 9.93T + 61T^{2} \) |
| 67 | \( 1 + 4.33T + 67T^{2} \) |
| 71 | \( 1 + 2.40T + 71T^{2} \) |
| 73 | \( 1 - 0.664T + 73T^{2} \) |
| 79 | \( 1 - 4.11T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 2.72T + 89T^{2} \) |
| 97 | \( 1 + 5.39T + 97T^{2} \) |
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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
−8.298183143505136369723256831784, −7.67092783960977655051848184201, −6.98432625091011843581074393297, −6.21717477134880380724555447469, −5.23389153704638988405430910067, −4.16630315273834277617163040257, −3.34595627729276658144384397504, −2.12177145077171970843276191175, −1.46206077749141236781653775166, 0,
1.46206077749141236781653775166, 2.12177145077171970843276191175, 3.34595627729276658144384397504, 4.16630315273834277617163040257, 5.23389153704638988405430910067, 6.21717477134880380724555447469, 6.98432625091011843581074393297, 7.67092783960977655051848184201, 8.298183143505136369723256831784