| L(s) = 1 | − 2·7-s + 3·11-s + 5·13-s + 6·17-s + 4·19-s + 3·23-s − 2·31-s + 11·37-s + 6·41-s + 4·43-s − 3·47-s − 3·49-s − 12·53-s − 9·59-s + 11·61-s − 14·67-s + 15·71-s + 2·73-s − 6·77-s + 10·79-s + 6·89-s − 10·91-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.904·11-s + 1.38·13-s + 1.45·17-s + 0.917·19-s + 0.625·23-s − 0.359·31-s + 1.80·37-s + 0.937·41-s + 0.609·43-s − 0.437·47-s − 3/7·49-s − 1.64·53-s − 1.17·59-s + 1.40·61-s − 1.71·67-s + 1.78·71-s + 0.234·73-s − 0.683·77-s + 1.12·79-s + 0.635·89-s − 1.04·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.639415758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.639415758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−16.49466495371778, −16.07353182842663, −15.50470541355798, −14.69041271610134, −14.28636863216231, −13.70324280723144, −13.01327631771580, −12.60182555941252, −11.90516701377239, −11.27450644729994, −10.83992938680892, −9.973449962204682, −9.282166160558584, −9.229302029070556, −8.048678783182206, −7.734356825328962, −6.786827683230665, −6.216980848591653, −5.759700306761675, −4.891976205663636, −3.920527814451729, −3.435000957965877, −2.787611932185244, −1.427805608297240, −0.8597377507637730,
0.8597377507637730, 1.427805608297240, 2.787611932185244, 3.435000957965877, 3.920527814451729, 4.891976205663636, 5.759700306761675, 6.216980848591653, 6.786827683230665, 7.734356825328962, 8.048678783182206, 9.229302029070556, 9.282166160558584, 9.973449962204682, 10.83992938680892, 11.27450644729994, 11.90516701377239, 12.60182555941252, 13.01327631771580, 13.70324280723144, 14.28636863216231, 14.69041271610134, 15.50470541355798, 16.07353182842663, 16.49466495371778
