| L(s) = 1 | + 4·7-s − 2·11-s + 13-s − 2·19-s + 2·23-s − 10·29-s − 4·31-s − 6·37-s + 6·41-s + 8·43-s − 12·47-s + 9·49-s + 14·53-s + 6·59-s + 2·61-s − 4·67-s + 14·73-s − 8·77-s + 12·83-s − 6·89-s + 4·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.603·11-s + 0.277·13-s − 0.458·19-s + 0.417·23-s − 1.85·29-s − 0.718·31-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 9/7·49-s + 1.92·53-s + 0.781·59-s + 0.256·61-s − 0.488·67-s + 1.63·73-s − 0.911·77-s + 1.31·83-s − 0.635·89-s + 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.627987360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.627987360\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.78056047732471, −13.37716009811676, −12.83611830084958, −12.43468971853959, −11.71598635701723, −11.29020438951187, −10.92091571343342, −10.57224041474802, −9.897405457732424, −9.216961600344466, −8.820705482864265, −8.267077054778050, −7.798597263763244, −7.362130113821909, −6.864196913382970, −6.028344620013419, −5.471984388164729, −5.150070310930314, −4.533613373445660, −3.888295203617379, −3.423637244610984, −2.376974634411125, −2.054064081493429, −1.355215304918368, −0.5187185028065239,
0.5187185028065239, 1.355215304918368, 2.054064081493429, 2.376974634411125, 3.423637244610984, 3.888295203617379, 4.533613373445660, 5.150070310930314, 5.471984388164729, 6.028344620013419, 6.864196913382970, 7.362130113821909, 7.798597263763244, 8.267077054778050, 8.820705482864265, 9.216961600344466, 9.897405457732424, 10.57224041474802, 10.92091571343342, 11.29020438951187, 11.71598635701723, 12.43468971853959, 12.83611830084958, 13.37716009811676, 13.78056047732471
