| L(s) = 1 | − 2·5-s − 11-s + 2·13-s − 17-s + 4·19-s − 25-s − 2·29-s − 8·31-s + 10·37-s + 6·41-s − 4·43-s − 8·47-s − 7·49-s + 6·53-s + 2·55-s − 12·59-s + 10·61-s − 4·65-s − 4·67-s − 16·71-s + 2·73-s − 8·79-s − 12·83-s + 2·85-s − 10·89-s − 8·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.301·11-s + 0.554·13-s − 0.242·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 1.64·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 49-s + 0.824·53-s + 0.269·55-s − 1.56·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s − 1.89·71-s + 0.234·73-s − 0.900·79-s − 1.31·83-s + 0.216·85-s − 1.05·89-s − 0.820·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9666713296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9666713296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.48748570960127, −13.20019897890027, −12.79187030764764, −12.15527523572590, −11.63557097457530, −11.23790953647103, −10.98584052011136, −10.27640773428407, −9.587409110315244, −9.405136086247811, −8.561537475256136, −8.256977692455075, −7.589458216706504, −7.361355117610930, −6.737000917723647, −5.939661587856389, −5.678817299793250, −4.923406798819896, −4.309115169381294, −3.906813699833186, −3.208992977912839, −2.810595230250828, −1.854185879775513, −1.253384646337988, −0.3175848924177828,
0.3175848924177828, 1.253384646337988, 1.854185879775513, 2.810595230250828, 3.208992977912839, 3.906813699833186, 4.309115169381294, 4.923406798819896, 5.678817299793250, 5.939661587856389, 6.737000917723647, 7.361355117610930, 7.589458216706504, 8.256977692455075, 8.561537475256136, 9.405136086247811, 9.587409110315244, 10.27640773428407, 10.98584052011136, 11.23790953647103, 11.63557097457530, 12.15527523572590, 12.79187030764764, 13.20019897890027, 13.48748570960127
