| L(s) = 1 | + 2·7-s + 6·11-s − 2·13-s + 6·17-s + 4·19-s + 8·23-s − 8·31-s + 2·37-s + 6·41-s + 4·43-s + 4·47-s − 3·49-s − 6·53-s + 6·59-s − 6·61-s + 4·71-s + 12·73-s + 12·77-s − 8·79-s − 12·83-s − 14·89-s − 4·91-s + 8·97-s − 8·101-s − 6·103-s − 12·107-s + 2·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.80·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.66·23-s − 1.43·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.824·53-s + 0.781·59-s − 0.768·61-s + 0.474·71-s + 1.40·73-s + 1.36·77-s − 0.900·79-s − 1.31·83-s − 1.48·89-s − 0.419·91-s + 0.812·97-s − 0.796·101-s − 0.591·103-s − 1.16·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.969468666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.969468666\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.77079584654901472480999050398, −7.29748649098476752106219363127, −6.65059647340821777600171379498, −5.67718320684817160469645811048, −5.18935084233229274182593774224, −4.31349073588253459211288439345, −3.59571571217919029122517285361, −2.79821995628023755071947183695, −1.53619986801494709991654485095, −1.00428492650276077651835595451,
1.00428492650276077651835595451, 1.53619986801494709991654485095, 2.79821995628023755071947183695, 3.59571571217919029122517285361, 4.31349073588253459211288439345, 5.18935084233229274182593774224, 5.67718320684817160469645811048, 6.65059647340821777600171379498, 7.29748649098476752106219363127, 7.77079584654901472480999050398
