L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s − 11-s + 2·15-s − 17-s + 2·19-s + 2·21-s + 2·23-s − 25-s − 27-s + 10·29-s + 8·31-s + 33-s + 4·35-s − 6·37-s + 2·41-s + 2·43-s − 2·45-s − 3·49-s + 51-s − 12·53-s + 2·55-s − 2·57-s + 10·59-s + 10·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.516·15-s − 0.242·17-s + 0.458·19-s + 0.436·21-s + 0.417·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.174·33-s + 0.676·35-s − 0.986·37-s + 0.312·41-s + 0.304·43-s − 0.298·45-s − 3/7·49-s + 0.140·51-s − 1.64·53-s + 0.269·55-s − 0.264·57-s + 1.30·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.065783668097754438215316323764, −7.06621691663350201741371213321, −6.65708475586121907062485079386, −5.82515585186678411877081006525, −4.93228111448097818803333957155, −4.28288740002267027276778366379, −3.37612124206513159950876073410, −2.61814430793643176061518948799, −1.08633931793910759593954775036, 0,
1.08633931793910759593954775036, 2.61814430793643176061518948799, 3.37612124206513159950876073410, 4.28288740002267027276778366379, 4.93228111448097818803333957155, 5.82515585186678411877081006525, 6.65708475586121907062485079386, 7.06621691663350201741371213321, 8.065783668097754438215316323764