| L(s) = 1 | + 5-s + 4·7-s + 11-s − 2·13-s − 17-s + 25-s − 2·29-s − 4·31-s + 4·35-s − 6·37-s + 2·41-s − 4·43-s − 8·47-s + 9·49-s + 14·53-s + 55-s + 4·59-s + 2·61-s − 2·65-s + 16·67-s − 12·71-s − 6·73-s + 4·77-s + 8·79-s − 12·83-s − 85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.301·11-s − 0.554·13-s − 0.242·17-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.676·35-s − 0.986·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.92·53-s + 0.134·55-s + 0.520·59-s + 0.256·61-s − 0.248·65-s + 1.95·67-s − 1.42·71-s − 0.702·73-s + 0.455·77-s + 0.900·79-s − 1.31·83-s − 0.108·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.76797196683054, −13.15696801986180, −12.86115613146906, −12.12096309854228, −11.69814877542542, −11.36797344743537, −10.87206130432541, −10.22422379820016, −10.00137664457002, −9.221891129777243, −8.759635397764989, −8.448820664150594, −7.742149007177880, −7.373271723362161, −6.830482238453772, −6.255235872480591, −5.538248200977489, −5.120380812919034, −4.812414299963839, −4.017697634663941, −3.621544235839763, −2.672081994217894, −2.127689144504851, −1.645254196778410, −1.017473168663113, 0,
1.017473168663113, 1.645254196778410, 2.127689144504851, 2.672081994217894, 3.621544235839763, 4.017697634663941, 4.812414299963839, 5.120380812919034, 5.538248200977489, 6.255235872480591, 6.830482238453772, 7.373271723362161, 7.742149007177880, 8.448820664150594, 8.759635397764989, 9.221891129777243, 10.00137664457002, 10.22422379820016, 10.87206130432541, 11.36797344743537, 11.69814877542542, 12.12096309854228, 12.86115613146906, 13.15696801986180, 13.76797196683054
