| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 4·11-s − 2·13-s + 15-s − 2·17-s − 4·19-s + 21-s + 23-s + 25-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 35-s − 2·37-s + 2·39-s + 2·41-s − 45-s − 4·47-s + 49-s + 2·51-s + 6·53-s + 4·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.169·35-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5441034010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5441034010\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.07300657213900, −14.32559433121431, −13.61999849442444, −13.23114559099882, −12.68250895326053, −12.22240396241990, −11.75612466874358, −11.12251043337547, −10.58887113547310, −10.15809743284049, −9.763651652239665, −8.811095819879676, −8.438192321731841, −7.811523592716668, −7.222658692036723, −6.642175896162881, −6.183747003300467, −5.424906733829048, −4.827735982854916, −4.417124818806420, −3.667307405024059, −2.687777224637560, −2.461267735757800, −1.245591680325796, −0.2958425280373538,
0.2958425280373538, 1.245591680325796, 2.461267735757800, 2.687777224637560, 3.667307405024059, 4.417124818806420, 4.827735982854916, 5.424906733829048, 6.183747003300467, 6.642175896162881, 7.222658692036723, 7.811523592716668, 8.438192321731841, 8.811095819879676, 9.763651652239665, 10.15809743284049, 10.58887113547310, 11.12251043337547, 11.75612466874358, 12.22240396241990, 12.68250895326053, 13.23114559099882, 13.61999849442444, 14.32559433121431, 15.07300657213900
