| L(s) = 1 | − 3-s + 7-s + 9-s + 3·11-s + 13-s + 3·17-s + 4·19-s − 21-s + 6·23-s − 27-s + 3·29-s + 31-s − 3·33-s + 2·37-s − 39-s + 10·43-s + 3·47-s − 6·49-s − 3·51-s − 3·53-s − 4·57-s − 3·59-s + 5·61-s + 63-s + 7·67-s − 6·69-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.727·17-s + 0.917·19-s − 0.218·21-s + 1.25·23-s − 0.192·27-s + 0.557·29-s + 0.179·31-s − 0.522·33-s + 0.328·37-s − 0.160·39-s + 1.52·43-s + 0.437·47-s − 6/7·49-s − 0.420·51-s − 0.412·53-s − 0.529·57-s − 0.390·59-s + 0.640·61-s + 0.125·63-s + 0.855·67-s − 0.722·69-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.564358269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.564358269\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−16.01747411512833, −15.56558144457136, −14.82562115810500, −14.33578905977122, −13.88893958760284, −13.16163774086806, −12.55282710739682, −12.01672883906565, −11.52807895833770, −10.99456276216660, −10.48438300236521, −9.651206402896603, −9.282800331685355, −8.586026804727022, −7.783745699799679, −7.342071664880025, −6.530109979957693, −6.111835526565547, −5.226476565075100, −4.867289170645177, −3.957120665853088, −3.352671655116296, −2.430135177968912, −1.288763029410627, −0.8513450854946882,
0.8513450854946882, 1.288763029410627, 2.430135177968912, 3.352671655116296, 3.957120665853088, 4.867289170645177, 5.226476565075100, 6.111835526565547, 6.530109979957693, 7.342071664880025, 7.783745699799679, 8.586026804727022, 9.282800331685355, 9.651206402896603, 10.48438300236521, 10.99456276216660, 11.52807895833770, 12.01672883906565, 12.55282710739682, 13.16163774086806, 13.88893958760284, 14.33578905977122, 14.82562115810500, 15.56558144457136, 16.01747411512833
