L(s) = 1 | + 7-s + 2·13-s − 6·17-s + 8·19-s + 6·29-s + 4·31-s − 10·37-s + 6·41-s + 4·43-s + 49-s + 6·53-s + 12·59-s + 10·61-s + 4·67-s + 12·71-s + 10·73-s − 8·79-s + 12·83-s + 6·89-s + 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·119-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 1.11·29-s + 0.718·31-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.56·59-s + 1.28·61-s + 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 1.31·83-s + 0.635·89-s + 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.437252845\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.437252845\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.72034005000020, −13.40946540132388, −12.79792550839560, −12.18934870429066, −11.78639111425988, −11.27091909306667, −10.94136561331743, −10.24951026146921, −9.879799813542370, −9.233602205554709, −8.681801316190183, −8.423656997091192, −7.703779888895508, −7.211650752449551, −6.660707658770153, −6.240431052219112, −5.416621215808470, −5.100825484711666, −4.492826947409687, −3.793808810733583, −3.370559186635375, −2.470453630491154, −2.119082647396861, −1.081965406257777, −0.6836987412878588,
0.6836987412878588, 1.081965406257777, 2.119082647396861, 2.470453630491154, 3.370559186635375, 3.793808810733583, 4.492826947409687, 5.100825484711666, 5.416621215808470, 6.240431052219112, 6.660707658770153, 7.211650752449551, 7.703779888895508, 8.423656997091192, 8.681801316190183, 9.233602205554709, 9.879799813542370, 10.24951026146921, 10.94136561331743, 11.27091909306667, 11.78639111425988, 12.18934870429066, 12.79792550839560, 13.40946540132388, 13.72034005000020