L(s) = 1 | − 3-s − 3·7-s + 9-s + 4.87·11-s + 2.87·13-s − 3.87·17-s − 2.87·19-s + 3·21-s + 23-s − 27-s − 5.87·29-s + 3·31-s − 4.87·33-s − 37-s − 2.87·39-s − 5.87·41-s − 3.74·43-s + 6.87·47-s + 2·49-s + 3.87·51-s + 3.87·53-s + 2.87·57-s + 5.87·59-s + 8.87·61-s − 3·63-s − 10.7·67-s − 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s + 0.333·9-s + 1.46·11-s + 0.796·13-s − 0.939·17-s − 0.659·19-s + 0.654·21-s + 0.208·23-s − 0.192·27-s − 1.09·29-s + 0.538·31-s − 0.848·33-s − 0.164·37-s − 0.460·39-s − 0.917·41-s − 0.571·43-s + 1.00·47-s + 0.285·49-s + 0.542·51-s + 0.531·53-s + 0.380·57-s + 0.764·59-s + 1.13·61-s − 0.377·63-s − 1.31·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 + 3.74T + 43T^{2} \) |
| 47 | \( 1 - 6.87T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 - 5.87T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 2.87T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 0.127T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.31372124712044032856701725429, −6.75000483164289383456751248925, −6.27787853369395932326921849596, −5.74492687759286602114812320301, −4.65691984422869507588858291033, −3.92252052516036106913815698297, −3.40200666873899334936974215853, −2.19091546781323659382449967471, −1.17600211755138191443109181817, 0,
1.17600211755138191443109181817, 2.19091546781323659382449967471, 3.40200666873899334936974215853, 3.92252052516036106913815698297, 4.65691984422869507588858291033, 5.74492687759286602114812320301, 6.27787853369395932326921849596, 6.75000483164289383456751248925, 7.31372124712044032856701725429