| L(s) = 1 | + 0.560·2-s − 3-s − 1.68·4-s − 0.560·6-s − 4.36·7-s − 2.06·8-s + 9-s − 3.72·11-s + 1.68·12-s + 2.53·13-s − 2.44·14-s + 2.21·16-s − 5.11·17-s + 0.560·18-s + 2.09·19-s + 4.36·21-s − 2.08·22-s − 6.47·23-s + 2.06·24-s + 1.41·26-s − 27-s + 7.35·28-s − 29-s − 9.94·31-s + 5.37·32-s + 3.72·33-s − 2.86·34-s + ⋯ |
| L(s) = 1 | + 0.396·2-s − 0.577·3-s − 0.843·4-s − 0.228·6-s − 1.64·7-s − 0.730·8-s + 0.333·9-s − 1.12·11-s + 0.486·12-s + 0.702·13-s − 0.653·14-s + 0.553·16-s − 1.24·17-s + 0.132·18-s + 0.481·19-s + 0.952·21-s − 0.445·22-s − 1.34·23-s + 0.421·24-s + 0.278·26-s − 0.192·27-s + 1.39·28-s − 0.185·29-s − 1.78·31-s + 0.949·32-s + 0.648·33-s − 0.491·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4758460027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4758460027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.560T + 2T^{2} \) |
| 7 | \( 1 + 4.36T + 7T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 + 5.11T + 17T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 31 | \( 1 + 9.94T + 31T^{2} \) |
| 37 | \( 1 + 5.37T + 37T^{2} \) |
| 41 | \( 1 - 7.96T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 9.38T + 47T^{2} \) |
| 53 | \( 1 - 5.64T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 9.81T + 61T^{2} \) |
| 67 | \( 1 + 0.520T + 67T^{2} \) |
| 71 | \( 1 - 3.22T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 1.81T + 79T^{2} \) |
| 83 | \( 1 - 1.35T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−9.247472353438144799923807195044, −8.393075183462618930749412685933, −7.39622507205374644593912608455, −6.48577776613506156291018596623, −5.81393679362756705500492572174, −5.23914833956844968650085705040, −4.06740684127135385280676452325, −3.54866664353302200250335193421, −2.38344205188089769363062483193, −0.41388092543812425358550677346,
0.41388092543812425358550677346, 2.38344205188089769363062483193, 3.54866664353302200250335193421, 4.06740684127135385280676452325, 5.23914833956844968650085705040, 5.81393679362756705500492572174, 6.48577776613506156291018596623, 7.39622507205374644593912608455, 8.393075183462618930749412685933, 9.247472353438144799923807195044
