| L(s) = 1 | + 1.92·3-s + 1.89·5-s − 1.20·7-s + 0.702·9-s + 11-s − 2.99·13-s + 3.64·15-s − 7.43·17-s + 5.74·19-s − 2.32·21-s − 3.07·23-s − 1.40·25-s − 4.42·27-s − 1.91·29-s − 0.0925·31-s + 1.92·33-s − 2.28·35-s − 8.35·37-s − 5.76·39-s − 1.05·41-s − 11.0·43-s + 1.33·45-s − 8.93·47-s − 5.54·49-s − 14.2·51-s + 0.142·53-s + 1.89·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 0.847·5-s − 0.456·7-s + 0.234·9-s + 0.301·11-s − 0.830·13-s + 0.941·15-s − 1.80·17-s + 1.31·19-s − 0.507·21-s − 0.640·23-s − 0.281·25-s − 0.850·27-s − 0.355·29-s − 0.0166·31-s + 0.334·33-s − 0.386·35-s − 1.37·37-s − 0.923·39-s − 0.165·41-s − 1.69·43-s + 0.198·45-s − 1.30·47-s − 0.791·49-s − 2.00·51-s + 0.0196·53-s + 0.255·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 1.92T + 3T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 13 | \( 1 + 2.99T + 13T^{2} \) |
| 17 | \( 1 + 7.43T + 17T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 + 3.07T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 + 0.0925T + 31T^{2} \) |
| 37 | \( 1 + 8.35T + 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 8.93T + 47T^{2} \) |
| 53 | \( 1 - 0.142T + 53T^{2} \) |
| 59 | \( 1 + 3.18T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 0.407T + 71T^{2} \) |
| 73 | \( 1 - 5.21T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 8.12T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 3.51T + 97T^{2} \) |
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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
−7.901383322057768711178957441795, −6.88459254201581680416992262703, −6.55868360039791724129199622638, −5.50353818917025331367435101671, −4.89043113648057299585681987381, −3.78741090051832405847156337156, −3.17768815375229227607137814715, −2.24732547520262490357182011271, −1.78764349193442259852712380916, 0,
1.78764349193442259852712380916, 2.24732547520262490357182011271, 3.17768815375229227607137814715, 3.78741090051832405847156337156, 4.89043113648057299585681987381, 5.50353818917025331367435101671, 6.55868360039791724129199622638, 6.88459254201581680416992262703, 7.901383322057768711178957441795
