| L(s) = 1 | + 5-s + 3·7-s + 3·11-s − 4·17-s + 6·19-s − 6·23-s − 4·25-s + 2·29-s + 9·31-s + 3·35-s − 2·37-s + 10·41-s + 6·43-s − 6·47-s + 2·49-s − 13·53-s + 3·55-s + 12·59-s + 8·61-s + 6·67-s − 12·71-s + 9·73-s + 9·77-s + 3·83-s − 4·85-s − 14·89-s + 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 0.904·11-s − 0.970·17-s + 1.37·19-s − 1.25·23-s − 4/5·25-s + 0.371·29-s + 1.61·31-s + 0.507·35-s − 0.328·37-s + 1.56·41-s + 0.914·43-s − 0.875·47-s + 2/7·49-s − 1.78·53-s + 0.404·55-s + 1.56·59-s + 1.02·61-s + 0.733·67-s − 1.42·71-s + 1.05·73-s + 1.02·77-s + 0.329·83-s − 0.433·85-s − 1.48·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.990665785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990665785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−10.00944520964452327346869373966, −9.431460471113781929238357373315, −8.407677972284915769308736818385, −7.75120135592916568614052087932, −6.65325213254225917768927617170, −5.81482520780945663550334444725, −4.78635677042327797457152601343, −3.94091986861202608778869492074, −2.42238256260612007081644598630, −1.30229225574123782841215729642,
1.30229225574123782841215729642, 2.42238256260612007081644598630, 3.94091986861202608778869492074, 4.78635677042327797457152601343, 5.81482520780945663550334444725, 6.65325213254225917768927617170, 7.75120135592916568614052087932, 8.407677972284915769308736818385, 9.431460471113781929238357373315, 10.00944520964452327346869373966
