| L(s) = 1 | + 1.93·3-s − 3.18·5-s − 4.42·7-s + 0.745·9-s + 11-s − 13-s − 6.17·15-s − 1.49·17-s − 1.76·19-s − 8.56·21-s + 1.61·23-s + 5.17·25-s − 4.36·27-s − 9.36·29-s − 5.31·31-s + 1.93·33-s + 14.1·35-s − 0.810·37-s − 1.93·39-s − 1.91·41-s + 5.23·43-s − 2.37·45-s + 8.37·47-s + 12.6·49-s − 2.88·51-s + 6.61·53-s − 3.18·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s − 1.42·5-s − 1.67·7-s + 0.248·9-s + 0.301·11-s − 0.277·13-s − 1.59·15-s − 0.361·17-s − 0.404·19-s − 1.86·21-s + 0.337·23-s + 1.03·25-s − 0.839·27-s − 1.73·29-s − 0.955·31-s + 0.336·33-s + 2.38·35-s − 0.133·37-s − 0.309·39-s − 0.299·41-s + 0.798·43-s − 0.354·45-s + 1.22·47-s + 1.80·49-s − 0.404·51-s + 0.908·53-s − 0.430·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 + 3.18T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 + 5.31T + 31T^{2} \) |
| 37 | \( 1 + 0.810T + 37T^{2} \) |
| 41 | \( 1 + 1.91T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 53 | \( 1 - 6.61T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 0.935T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 3.18T + 97T^{2} \) |
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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
−10.13870434593571344712369698311, −9.111649472455917210476613530981, −8.750924507905830603109157826616, −7.52260326081183085214054178546, −7.06957375615831871973604329480, −5.77502881856884976855256136452, −4.01751963939822849215335786609, −3.57572587212974432305644081881, −2.53721637069412439813848991395, 0,
2.53721637069412439813848991395, 3.57572587212974432305644081881, 4.01751963939822849215335786609, 5.77502881856884976855256136452, 7.06957375615831871973604329480, 7.52260326081183085214054178546, 8.750924507905830603109157826616, 9.111649472455917210476613530981, 10.13870434593571344712369698311
