| L(s) = 1 | − 2.82·3-s + 1.41·5-s + 5.00·9-s + 11-s − 1.41·13-s − 4.00·15-s + 7.07·17-s − 2.82·19-s − 4·23-s − 2.99·25-s − 5.65·27-s + 5.65·31-s − 2.82·33-s − 8·37-s + 4.00·39-s + 9.89·41-s − 4·43-s + 7.07·45-s − 20.0·51-s − 6·53-s + 1.41·55-s + 8.00·57-s − 8.48·59-s − 1.41·61-s − 2.00·65-s + 8·67-s + 11.3·69-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 0.632·5-s + 1.66·9-s + 0.301·11-s − 0.392·13-s − 1.03·15-s + 1.71·17-s − 0.648·19-s − 0.834·23-s − 0.599·25-s − 1.08·27-s + 1.01·31-s − 0.492·33-s − 1.31·37-s + 0.640·39-s + 1.54·41-s − 0.609·43-s + 1.05·45-s − 2.80·51-s − 0.824·53-s + 0.190·55-s + 1.05·57-s − 1.10·59-s − 0.181·61-s − 0.248·65-s + 0.977·67-s + 1.36·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−7.29539316156711661568897171555, −6.47726598872353544502337158923, −5.95446841228358925288734060977, −5.56911407749023556267444126448, −4.79348895076068348219361886611, −4.13480041904322698132155791797, −3.10662169369221883707936802326, −1.91441368626081719640283111893, −1.12113155743030734812238244327, 0,
1.12113155743030734812238244327, 1.91441368626081719640283111893, 3.10662169369221883707936802326, 4.13480041904322698132155791797, 4.79348895076068348219361886611, 5.56911407749023556267444126448, 5.95446841228358925288734060977, 6.47726598872353544502337158923, 7.29539316156711661568897171555
