| L(s) = 1 | − 2·3-s + 4·7-s + 9-s − 4·11-s − 6·17-s − 19-s − 8·21-s + 8·23-s + 4·27-s + 6·29-s + 8·31-s + 8·33-s − 8·37-s − 2·41-s + 12·47-s + 9·49-s + 12·51-s + 4·53-s + 2·57-s + 8·59-s + 14·61-s + 4·63-s + 2·67-s − 16·69-s + 8·71-s + 2·73-s − 16·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.45·17-s − 0.229·19-s − 1.74·21-s + 1.66·23-s + 0.769·27-s + 1.11·29-s + 1.43·31-s + 1.39·33-s − 1.31·37-s − 0.312·41-s + 1.75·47-s + 9/7·49-s + 1.68·51-s + 0.549·53-s + 0.264·57-s + 1.04·59-s + 1.79·61-s + 0.503·63-s + 0.244·67-s − 1.92·69-s + 0.949·71-s + 0.234·73-s − 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496748140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496748140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.22934966826715, −14.68045802889406, −13.96447002558351, −13.56890889298405, −12.93244899318707, −12.37143314868904, −11.77345460047309, −11.31605365617444, −10.95687275482365, −10.45796792803621, −10.09117215905124, −8.931132832020347, −8.517541601271346, −8.184792186392194, −7.221882457255229, −6.880326802719049, −6.206816251676718, −5.325260101087849, −5.100107969235846, −4.671802436180239, −3.951134489846551, −2.682373865780311, −2.354492561250410, −1.234865372918411, −0.5558085086232145,
0.5558085086232145, 1.234865372918411, 2.354492561250410, 2.682373865780311, 3.951134489846551, 4.671802436180239, 5.100107969235846, 5.325260101087849, 6.206816251676718, 6.880326802719049, 7.221882457255229, 8.184792186392194, 8.517541601271346, 8.931132832020347, 10.09117215905124, 10.45796792803621, 10.95687275482365, 11.31605365617444, 11.77345460047309, 12.37143314868904, 12.93244899318707, 13.56890889298405, 13.96447002558351, 14.68045802889406, 15.22934966826715
