L(s) = 1 | + 3-s − 2·5-s − 2·7-s + 9-s − 11-s + 4·13-s − 2·15-s − 2·19-s − 2·21-s + 2·23-s − 25-s + 27-s − 2·29-s − 4·31-s − 33-s + 4·35-s − 6·37-s + 4·39-s − 6·41-s − 2·43-s − 2·45-s − 3·49-s − 12·53-s + 2·55-s − 2·57-s + 14·59-s − 6·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.516·15-s − 0.458·19-s − 0.436·21-s + 0.417·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s + 0.676·35-s − 0.986·37-s + 0.640·39-s − 0.937·41-s − 0.304·43-s − 0.298·45-s − 3/7·49-s − 1.64·53-s + 0.269·55-s − 0.264·57-s + 1.82·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.44779554601698, −13.15265856231723, −12.67070158423406, −12.22748366392301, −11.64256474022097, −11.14875775570126, −10.75310568917392, −10.22836811748187, −9.641428327750274, −9.221324025209762, −8.606521474023265, −8.320034523168385, −7.810487357472680, −7.250498401837000, −6.762915873281567, −6.299218692995968, −5.696425893286019, −5.014556506600114, −4.481275241413333, −3.698330548041957, −3.503800729226088, −3.104804879209079, −2.150880777714445, −1.679317809856498, −0.7170164052490554, 0,
0.7170164052490554, 1.679317809856498, 2.150880777714445, 3.104804879209079, 3.503800729226088, 3.698330548041957, 4.481275241413333, 5.014556506600114, 5.696425893286019, 6.299218692995968, 6.762915873281567, 7.250498401837000, 7.810487357472680, 8.320034523168385, 8.606521474023265, 9.221324025209762, 9.641428327750274, 10.22836811748187, 10.75310568917392, 11.14875775570126, 11.64256474022097, 12.22748366392301, 12.67070158423406, 13.15265856231723, 13.44779554601698