L(s) = 1 | + 2.83·3-s − 2.22·5-s + 5.01·9-s − 1.51·11-s + 6.85·13-s − 6.28·15-s + 1.59·17-s − 2.36·19-s + 4.25·23-s − 0.0656·25-s + 5.70·27-s + 1.45·29-s + 2.09·31-s − 4.29·33-s + 7.96·37-s + 19.3·39-s − 41-s + 3.59·43-s − 11.1·45-s − 6.92·47-s + 4.51·51-s − 7.14·53-s + 3.36·55-s − 6.68·57-s − 3.14·59-s + 3.64·61-s − 15.2·65-s + ⋯ |
L(s) = 1 | + 1.63·3-s − 0.993·5-s + 1.67·9-s − 0.456·11-s + 1.89·13-s − 1.62·15-s + 0.386·17-s − 0.542·19-s + 0.887·23-s − 0.0131·25-s + 1.09·27-s + 0.269·29-s + 0.375·31-s − 0.746·33-s + 1.30·37-s + 3.10·39-s − 0.156·41-s + 0.547·43-s − 1.66·45-s − 1.01·47-s + 0.632·51-s − 0.982·53-s + 0.453·55-s − 0.886·57-s − 0.409·59-s + 0.466·61-s − 1.88·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.562821008\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.562821008\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 2.83T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 - 6.85T + 13T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 + 2.36T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 - 7.96T + 37T^{2} \) |
| 43 | \( 1 - 3.59T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 + 3.14T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 - 8.76T + 67T^{2} \) |
| 71 | \( 1 + 0.569T + 71T^{2} \) |
| 73 | \( 1 - 3.79T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 5.82T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.061242845401808110411936825255, −7.44968764560712790085684324288, −6.62110625845797429480657681378, −5.86408143316187524275388353037, −4.68467388279131935178503105122, −4.04826620248799991958500981102, −3.40525829846169667473339797456, −2.93786370252781191114343014132, −1.89945334566780144392103800330, −0.891164744877721232594817443471,
0.891164744877721232594817443471, 1.89945334566780144392103800330, 2.93786370252781191114343014132, 3.40525829846169667473339797456, 4.04826620248799991958500981102, 4.68467388279131935178503105122, 5.86408143316187524275388353037, 6.62110625845797429480657681378, 7.44968764560712790085684324288, 8.061242845401808110411936825255