| L(s) = 1 | + 1.46·2-s + 3-s + 0.150·4-s + 1.27·5-s + 1.46·6-s − 2.71·8-s + 9-s + 1.87·10-s + 4.55·11-s + 0.150·12-s − 3.82·13-s + 1.27·15-s − 4.27·16-s − 3.03·17-s + 1.46·18-s − 4.80·19-s + 0.192·20-s + 6.68·22-s − 6.82·23-s − 2.71·24-s − 3.36·25-s − 5.61·26-s + 27-s − 8.73·29-s + 1.87·30-s + 0.252·31-s − 0.850·32-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 0.577·3-s + 0.0752·4-s + 0.572·5-s + 0.598·6-s − 0.958·8-s + 0.333·9-s + 0.593·10-s + 1.37·11-s + 0.0434·12-s − 1.06·13-s + 0.330·15-s − 1.06·16-s − 0.735·17-s + 0.345·18-s − 1.10·19-s + 0.0430·20-s + 1.42·22-s − 1.42·23-s − 0.553·24-s − 0.672·25-s − 1.10·26-s + 0.192·27-s − 1.62·29-s + 0.342·30-s + 0.0454·31-s − 0.150·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 5 | \( 1 - 1.27T + 5T^{2} \) |
| 11 | \( 1 - 4.55T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 + 4.80T + 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 + 8.73T + 29T^{2} \) |
| 31 | \( 1 - 0.252T + 31T^{2} \) |
| 37 | \( 1 - 1.57T + 37T^{2} \) |
| 43 | \( 1 - 3.30T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 - 3.70T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 9.19T + 67T^{2} \) |
| 71 | \( 1 + 6.31T + 71T^{2} \) |
| 73 | \( 1 - 1.54T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 6.14T + 89T^{2} \) |
| 97 | \( 1 + 9.51T + 97T^{2} \) |
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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
−7.63805492985758048518882711282, −6.80560848840179019892265357311, −6.14309870525693544462015531277, −5.63096556747021810494934397338, −4.52366441827032237902394625328, −4.17158280515814694325747270315, −3.44904737615971078076009493270, −2.34933373930619331690958084732, −1.83112117967587843631971173589, 0,
1.83112117967587843631971173589, 2.34933373930619331690958084732, 3.44904737615971078076009493270, 4.17158280515814694325747270315, 4.52366441827032237902394625328, 5.63096556747021810494934397338, 6.14309870525693544462015531277, 6.80560848840179019892265357311, 7.63805492985758048518882711282
