L(s) = 1 | + 5-s − 7-s + 2·11-s + 13-s − 4·17-s + 8·19-s − 8·23-s + 25-s + 6·29-s + 2·31-s − 35-s + 10·37-s − 4·41-s − 2·43-s − 4·47-s + 49-s + 2·55-s + 2·61-s + 65-s − 12·67-s − 8·71-s + 2·73-s − 2·77-s − 4·83-s − 4·85-s + 12·89-s − 91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.603·11-s + 0.277·13-s − 0.970·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s − 0.169·35-s + 1.64·37-s − 0.624·41-s − 0.304·43-s − 0.583·47-s + 1/7·49-s + 0.269·55-s + 0.256·61-s + 0.124·65-s − 1.46·67-s − 0.949·71-s + 0.234·73-s − 0.227·77-s − 0.439·83-s − 0.433·85-s + 1.27·89-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.44930896374137, −13.82612964352906, −13.49012498727031, −13.19972067909747, −12.30589202415030, −11.94937294375673, −11.56098499566257, −10.96789546242595, −10.22421288621707, −9.918885945102556, −9.404048183782208, −8.953104852104338, −8.277650335809688, −7.804976266174506, −7.162018492720807, −6.447210386999670, −6.253327486866141, −5.585811319326835, −4.927686917085970, −4.291105591734489, −3.761329606622257, −2.980234096567375, −2.512334334452478, −1.602030186576478, −1.044490738471343, 0,
1.044490738471343, 1.602030186576478, 2.512334334452478, 2.980234096567375, 3.761329606622257, 4.291105591734489, 4.927686917085970, 5.585811319326835, 6.253327486866141, 6.447210386999670, 7.162018492720807, 7.804976266174506, 8.277650335809688, 8.953104852104338, 9.404048183782208, 9.918885945102556, 10.22421288621707, 10.96789546242595, 11.56098499566257, 11.94937294375673, 12.30589202415030, 13.19972067909747, 13.49012498727031, 13.82612964352906, 14.44930896374137