L(s) = 1 | − 2.36·3-s − 5-s − 4.12·7-s + 2.60·9-s − 5.68·11-s + 5.13·13-s + 2.36·15-s − 0.970·17-s + 4.34·19-s + 9.78·21-s − 2.71·23-s + 25-s + 0.926·27-s + 3.21·29-s − 5.47·31-s + 13.4·33-s + 4.12·35-s − 4.24·37-s − 12.1·39-s − 7.94·41-s − 10.2·43-s − 2.60·45-s − 7.77·47-s + 10.0·49-s + 2.29·51-s − 1.02·53-s + 5.68·55-s + ⋯ |
L(s) = 1 | − 1.36·3-s − 0.447·5-s − 1.56·7-s + 0.869·9-s − 1.71·11-s + 1.42·13-s + 0.611·15-s − 0.235·17-s + 0.995·19-s + 2.13·21-s − 0.565·23-s + 0.200·25-s + 0.178·27-s + 0.597·29-s − 0.982·31-s + 2.34·33-s + 0.698·35-s − 0.697·37-s − 1.94·39-s − 1.24·41-s − 1.56·43-s − 0.388·45-s − 1.13·47-s + 1.43·49-s + 0.321·51-s − 0.141·53-s + 0.767·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08449090427\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08449090427\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 2.36T + 3T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 11 | \( 1 + 5.68T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 + 0.970T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 7.94T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 7.77T + 47T^{2} \) |
| 53 | \( 1 + 1.02T + 53T^{2} \) |
| 59 | \( 1 + 0.108T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 3.21T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 8.14T + 79T^{2} \) |
| 83 | \( 1 - 3.54T + 83T^{2} \) |
| 89 | \( 1 - 2.64T + 89T^{2} \) |
| 97 | \( 1 + 4.33T + 97T^{2} \) |
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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
−7.76325949365738645484996230327, −6.82178193964287375269483450241, −6.50281040146257490482736009923, −5.65344975497610542019104933485, −5.33082179128923802329107355757, −4.41157243554513393589356919906, −3.34892293273494719967908364290, −3.02253745636608522876937330891, −1.48998697583644040636310863044, −0.15605810224658256626552932406,
0.15605810224658256626552932406, 1.48998697583644040636310863044, 3.02253745636608522876937330891, 3.34892293273494719967908364290, 4.41157243554513393589356919906, 5.33082179128923802329107355757, 5.65344975497610542019104933485, 6.50281040146257490482736009923, 6.82178193964287375269483450241, 7.76325949365738645484996230327