L(s) = 1 | + 2·3-s − 4·7-s + 9-s + 11-s + 4·19-s − 8·21-s + 6·23-s − 4·27-s − 6·29-s − 8·31-s + 2·33-s + 2·37-s − 6·41-s − 8·43-s + 6·47-s + 9·49-s + 6·53-s + 8·57-s + 12·59-s + 2·61-s − 4·63-s − 10·67-s + 12·69-s + 12·71-s − 16·73-s − 4·77-s + 8·79-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.917·19-s − 1.74·21-s + 1.25·23-s − 0.769·27-s − 1.11·29-s − 1.43·31-s + 0.348·33-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s + 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.503·63-s − 1.22·67-s + 1.44·69-s + 1.42·71-s − 1.87·73-s − 0.455·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.32708354134553, −13.16726571423253, −12.57161824129951, −12.02065399353313, −11.52629806510504, −11.00305395281972, −10.42857380708933, −9.718958106138216, −9.644304219818891, −9.046763856981296, −8.779021391935911, −8.234052472387562, −7.510796887044312, −7.071324652861625, −6.893855279504118, −6.054926156328935, −5.565414940660799, −5.154283023387349, −4.205932450645074, −3.694959885938043, −3.277349743659492, −2.989024553122660, −2.266737010856106, −1.686078510723295, −0.8038805660167673, 0,
0.8038805660167673, 1.686078510723295, 2.266737010856106, 2.989024553122660, 3.277349743659492, 3.694959885938043, 4.205932450645074, 5.154283023387349, 5.565414940660799, 6.054926156328935, 6.893855279504118, 7.071324652861625, 7.510796887044312, 8.234052472387562, 8.779021391935911, 9.046763856981296, 9.644304219818891, 9.718958106138216, 10.42857380708933, 11.00305395281972, 11.52629806510504, 12.02065399353313, 12.57161824129951, 13.16726571423253, 13.32708354134553