L(s) = 1 | + 2·7-s − 11-s − 2·13-s + 17-s + 4·19-s + 6·23-s − 5·25-s + 4·29-s + 2·31-s − 4·37-s + 2·41-s + 4·43-s − 3·49-s − 2·53-s + 4·59-s − 12·67-s + 2·71-s + 2·73-s − 2·77-s + 14·79-s + 12·83-s − 6·89-s − 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 0.301·11-s − 0.554·13-s + 0.242·17-s + 0.917·19-s + 1.25·23-s − 25-s + 0.742·29-s + 0.359·31-s − 0.657·37-s + 0.312·41-s + 0.609·43-s − 3/7·49-s − 0.274·53-s + 0.520·59-s − 1.46·67-s + 0.237·71-s + 0.234·73-s − 0.227·77-s + 1.57·79-s + 1.31·83-s − 0.635·89-s − 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.506081977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.506081977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.08089406873507, −14.90847024193202, −14.11332625028846, −13.77140431722442, −13.24765125784138, −12.42910585142795, −12.13027752856958, −11.47730930933530, −11.01384399848667, −10.43263798826051, −9.797568809787115, −9.337970975033255, −8.643552567932641, −8.045974159160313, −7.540129381807662, −7.072832395817686, −6.281292778870555, −5.587520900837342, −4.975274403979943, −4.619612438530123, −3.680230805875750, −3.008583752348152, −2.297292786448140, −1.465268358968674, −0.6455380053171409,
0.6455380053171409, 1.465268358968674, 2.297292786448140, 3.008583752348152, 3.680230805875750, 4.619612438530123, 4.975274403979943, 5.587520900837342, 6.281292778870555, 7.072832395817686, 7.540129381807662, 8.045974159160313, 8.643552567932641, 9.337970975033255, 9.797568809787115, 10.43263798826051, 11.01384399848667, 11.47730930933530, 12.13027752856958, 12.42910585142795, 13.24765125784138, 13.77140431722442, 14.11332625028846, 14.90847024193202, 15.08089406873507