| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s + 6·13-s − 15-s − 7·17-s − 5·19-s − 21-s + 23-s + 25-s + 27-s + 5·29-s + 8·31-s + 33-s + 35-s + 2·37-s + 6·39-s + 12·41-s − 11·43-s − 45-s − 8·47-s + 49-s − 7·51-s + 11·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s − 0.258·15-s − 1.69·17-s − 1.14·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.928·29-s + 1.43·31-s + 0.174·33-s + 0.169·35-s + 0.328·37-s + 0.960·39-s + 1.87·41-s − 1.67·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.980·51-s + 1.51·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.574200945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.574200945\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.89897895285660, −13.54208489012480, −13.18291099878485, −12.78657710710460, −12.00884497105837, −11.63409995007930, −10.93248938244291, −10.70041484697667, −10.09916517407432, −9.363357062048350, −8.872971121836718, −8.515350124834886, −8.166019555311001, −7.451841852084630, −6.663611587831652, −6.394282385515099, −6.065385031442509, −4.932721825326802, −4.377711015943798, −4.055892039982798, −3.350526301028488, −2.769818099556902, −2.109012942409227, −1.317452386631707, −0.5303162762638641,
0.5303162762638641, 1.317452386631707, 2.109012942409227, 2.769818099556902, 3.350526301028488, 4.055892039982798, 4.377711015943798, 4.932721825326802, 6.065385031442509, 6.394282385515099, 6.663611587831652, 7.451841852084630, 8.166019555311001, 8.515350124834886, 8.872971121836718, 9.363357062048350, 10.09916517407432, 10.70041484697667, 10.93248938244291, 11.63409995007930, 12.00884497105837, 12.78657710710460, 13.18291099878485, 13.54208489012480, 13.89897895285660
