L(s) = 1 | − 3-s − 2·7-s + 9-s − 2·11-s − 2·13-s + 4·17-s − 6·19-s + 2·21-s + 6·23-s − 27-s + 10·29-s + 8·31-s + 2·33-s + 2·37-s + 2·39-s + 2·41-s + 43-s − 2·47-s − 3·49-s − 4·51-s + 10·53-s + 6·57-s + 2·59-s + 12·61-s − 2·63-s + 12·67-s − 6·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.348·33-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 0.152·43-s − 0.291·47-s − 3/7·49-s − 0.560·51-s + 1.37·53-s + 0.794·57-s + 0.260·59-s + 1.53·61-s − 0.251·63-s + 1.46·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.17188701774151, −12.79360427234135, −12.34828252878507, −11.86817602543513, −11.55122757253248, −10.78203198442341, −10.31728538097200, −10.21532291869545, −9.634184012595651, −9.064321971007108, −8.450362614530631, −8.111890601421689, −7.478262901270287, −6.917923609870027, −6.475288370732130, −6.195250062402587, −5.401043386717450, −5.074669037171773, −4.498154007236609, −4.000826002573458, −3.246128573745394, −2.646046719874540, −2.397393028341198, −1.242786876606136, −0.7802633924337233, 0,
0.7802633924337233, 1.242786876606136, 2.397393028341198, 2.646046719874540, 3.246128573745394, 4.000826002573458, 4.498154007236609, 5.074669037171773, 5.401043386717450, 6.195250062402587, 6.475288370732130, 6.917923609870027, 7.478262901270287, 8.111890601421689, 8.450362614530631, 9.064321971007108, 9.634184012595651, 10.21532291869545, 10.31728538097200, 10.78203198442341, 11.55122757253248, 11.86817602543513, 12.34828252878507, 12.79360427234135, 13.17188701774151