| L(s) = 1 | − 2·5-s + 7-s − 2·11-s + 6·17-s + 4·19-s + 6·23-s − 25-s + 4·31-s − 2·35-s − 10·37-s − 2·41-s − 4·43-s − 4·47-s + 49-s − 12·53-s + 4·55-s − 12·59-s + 6·61-s + 4·67-s + 14·71-s + 2·73-s − 2·77-s − 8·79-s + 16·83-s − 12·85-s + 6·89-s − 8·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.603·11-s + 1.45·17-s + 0.917·19-s + 1.25·23-s − 1/5·25-s + 0.718·31-s − 0.338·35-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 1.64·53-s + 0.539·55-s − 1.56·59-s + 0.768·61-s + 0.488·67-s + 1.66·71-s + 0.234·73-s − 0.227·77-s − 0.900·79-s + 1.75·83-s − 1.30·85-s + 0.635·89-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.884233977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.884233977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.93909690827556, −13.51662521840736, −12.82698087164659, −12.39697861131477, −11.86467300971944, −11.60074918655780, −10.95593126133060, −10.51831919169441, −9.961461308385483, −9.446063711335826, −8.874618442107424, −8.084569871983846, −7.969106014598110, −7.456819297360258, −6.870602680110677, −6.284380576266757, −5.462203777580945, −5.025375815953033, −4.736636506480263, −3.703421830501146, −3.370814987117096, −2.899148971767708, −1.915937875457879, −1.214593326788762, −0.4819601724126197,
0.4819601724126197, 1.214593326788762, 1.915937875457879, 2.899148971767708, 3.370814987117096, 3.703421830501146, 4.736636506480263, 5.025375815953033, 5.462203777580945, 6.284380576266757, 6.870602680110677, 7.456819297360258, 7.969106014598110, 8.084569871983846, 8.874618442107424, 9.446063711335826, 9.961461308385483, 10.51831919169441, 10.95593126133060, 11.60074918655780, 11.86467300971944, 12.39697861131477, 12.82698087164659, 13.51662521840736, 13.93909690827556
