| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 11-s + 2·13-s + 14-s + 16-s − 2·17-s + 6·19-s + 20-s + 22-s − 6·23-s + 25-s + 2·26-s + 28-s − 4·29-s + 32-s − 2·34-s + 35-s + 8·37-s + 6·38-s + 40-s + 4·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.742·29-s + 0.176·32-s − 0.342·34-s + 0.169·35-s + 1.31·37-s + 0.973·38-s + 0.158·40-s + 0.609·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.150686487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.150686487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.80796760467972155805936780007, −7.21681494747971363289924327398, −6.37511435337672681377273489710, −5.76712477873642013489332403986, −5.23987002325723155733684807338, −4.26366267078329965014242866489, −3.75985904836752477488695564835, −2.73525220513410119007818600834, −1.95051448669355241985414253037, −0.979934963746224700502343421621,
0.979934963746224700502343421621, 1.95051448669355241985414253037, 2.73525220513410119007818600834, 3.75985904836752477488695564835, 4.26366267078329965014242866489, 5.23987002325723155733684807338, 5.76712477873642013489332403986, 6.37511435337672681377273489710, 7.21681494747971363289924327398, 7.80796760467972155805936780007
