| L(s) = 1 | + 3-s + 1.90·5-s + 2.67·7-s + 9-s + 2.07·11-s − 4.12·13-s + 1.90·15-s + 7.22·17-s + 5.16·19-s + 2.67·21-s − 23-s − 1.37·25-s + 27-s + 29-s − 0.246·31-s + 2.07·33-s + 5.08·35-s + 3.33·37-s − 4.12·39-s − 8.46·41-s + 10.5·43-s + 1.90·45-s − 7.13·47-s + 0.139·49-s + 7.22·51-s + 5.50·53-s + 3.94·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.851·5-s + 1.00·7-s + 0.333·9-s + 0.625·11-s − 1.14·13-s + 0.491·15-s + 1.75·17-s + 1.18·19-s + 0.583·21-s − 0.208·23-s − 0.275·25-s + 0.192·27-s + 0.185·29-s − 0.0443·31-s + 0.361·33-s + 0.859·35-s + 0.547·37-s − 0.660·39-s − 1.32·41-s + 1.60·43-s + 0.283·45-s − 1.04·47-s + 0.0199·49-s + 1.01·51-s + 0.756·53-s + 0.532·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.026630533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.026630533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 1.90T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 31 | \( 1 + 0.246T + 31T^{2} \) |
| 37 | \( 1 - 3.33T + 37T^{2} \) |
| 41 | \( 1 + 8.46T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 7.13T + 47T^{2} \) |
| 53 | \( 1 - 5.50T + 53T^{2} \) |
| 59 | \( 1 - 6.97T + 59T^{2} \) |
| 61 | \( 1 + 6.60T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 3.00T + 73T^{2} \) |
| 79 | \( 1 + 6.72T + 79T^{2} \) |
| 83 | \( 1 - 4.78T + 83T^{2} \) |
| 89 | \( 1 + 9.34T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.82441188060222723550542433682, −7.34156077690622604704027447507, −6.50568689342153769143110738135, −5.49060372096647750672098931629, −5.23891653914183245332136408245, −4.29552263386548456038936650862, −3.42180026164802775059805115781, −2.60189642994563625989314949564, −1.76053833418446292805396035998, −1.06185176282534005715583342360,
1.06185176282534005715583342360, 1.76053833418446292805396035998, 2.60189642994563625989314949564, 3.42180026164802775059805115781, 4.29552263386548456038936650862, 5.23891653914183245332136408245, 5.49060372096647750672098931629, 6.50568689342153769143110738135, 7.34156077690622604704027447507, 7.82441188060222723550542433682
