| L(s) = 1 | − 3-s + 4·7-s + 9-s + 4·11-s + 4·17-s − 4·21-s + 4·23-s − 27-s + 6·29-s + 4·31-s − 4·33-s + 8·37-s − 10·41-s − 4·43-s − 4·47-s + 9·49-s − 4·51-s + 12·53-s − 4·59-s − 2·61-s + 4·63-s + 4·67-s − 4·69-s − 8·73-s + 16·77-s − 12·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.970·17-s − 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.696·33-s + 1.31·37-s − 1.56·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s − 0.560·51-s + 1.64·53-s − 0.520·59-s − 0.256·61-s + 0.503·63-s + 0.488·67-s − 0.481·69-s − 0.936·73-s + 1.82·77-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.407730990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.407730990\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.336507137889844401891980847807, −7.53276757494499987418620340530, −6.82457173259429192457012851926, −6.09942867346529735042589671382, −5.23700493314447416785122665215, −4.68579410363484290514444551060, −3.96598999890927710316427824366, −2.86734108346247982598876834498, −1.57514137059454685314359201386, −1.01521734459139859012121055491,
1.01521734459139859012121055491, 1.57514137059454685314359201386, 2.86734108346247982598876834498, 3.96598999890927710316427824366, 4.68579410363484290514444551060, 5.23700493314447416785122665215, 6.09942867346529735042589671382, 6.82457173259429192457012851926, 7.53276757494499987418620340530, 8.336507137889844401891980847807
