| L(s) = 1 | − 2-s + 4-s + 4·5-s − 8-s − 4·10-s + 4·11-s − 3·13-s + 16-s − 7·17-s − 2·19-s + 4·20-s − 4·22-s + 23-s + 11·25-s + 3·26-s − 29-s + 9·31-s − 32-s + 7·34-s + 2·37-s + 2·38-s − 4·40-s + 6·41-s + 11·43-s + 4·44-s − 46-s − 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 1.20·11-s − 0.832·13-s + 1/4·16-s − 1.69·17-s − 0.458·19-s + 0.894·20-s − 0.852·22-s + 0.208·23-s + 11/5·25-s + 0.588·26-s − 0.185·29-s + 1.61·31-s − 0.176·32-s + 1.20·34-s + 0.328·37-s + 0.324·38-s − 0.632·40-s + 0.937·41-s + 1.67·43-s + 0.603·44-s − 0.147·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909344415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909344415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−9.076386420504345035537011996118, −8.386834803015030063906818806717, −7.17230892063836621876590463825, −6.49244247951320148958731343293, −6.12230164783542580596662547870, −5.05352817881679124240654755585, −4.16226096911516495331985392522, −2.57898691887524846597709635063, −2.13967823750617304276182350687, −0.998858625839110032845229095752,
0.998858625839110032845229095752, 2.13967823750617304276182350687, 2.57898691887524846597709635063, 4.16226096911516495331985392522, 5.05352817881679124240654755585, 6.12230164783542580596662547870, 6.49244247951320148958731343293, 7.17230892063836621876590463825, 8.386834803015030063906818806717, 9.076386420504345035537011996118
