| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 3·11-s − 15-s − 2·17-s − 7·19-s − 21-s − 23-s + 25-s + 27-s − 2·29-s + 4·31-s + 3·33-s + 35-s + 2·37-s + 3·41-s − 10·43-s − 45-s + 47-s + 49-s − 2·51-s − 3·53-s − 3·55-s − 7·57-s + 5·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.258·15-s − 0.485·17-s − 1.60·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.522·33-s + 0.169·35-s + 0.328·37-s + 0.468·41-s − 1.52·43-s − 0.149·45-s + 0.145·47-s + 1/7·49-s − 0.280·51-s − 0.412·53-s − 0.404·55-s − 0.927·57-s + 0.650·59-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.09728925294630, −14.59516643811846, −14.13228343777817, −13.51858545935742, −12.96792033866215, −12.60553389444575, −11.97946354361767, −11.41497128463919, −10.94095213621265, −10.28030724436866, −9.723266549924363, −9.239605449423020, −8.518241484395030, −8.360392904271110, −7.614118826624403, −6.844127774693815, −6.558356978972782, −5.980259231762896, −5.016166391095922, −4.401016699048758, −3.865191487783462, −3.400963153287989, −2.479044613522647, −1.975042849451093, −0.9997167112780474, 0,
0.9997167112780474, 1.975042849451093, 2.479044613522647, 3.400963153287989, 3.865191487783462, 4.401016699048758, 5.016166391095922, 5.980259231762896, 6.558356978972782, 6.844127774693815, 7.614118826624403, 8.360392904271110, 8.518241484395030, 9.239605449423020, 9.723266549924363, 10.28030724436866, 10.94095213621265, 11.41497128463919, 11.97946354361767, 12.60553389444575, 12.96792033866215, 13.51858545935742, 14.13228343777817, 14.59516643811846, 15.09728925294630
