L(s) = 1 | + 0.277·3-s + 5-s − 7-s − 2.92·9-s + 4·11-s − 13-s + 0.277·15-s + 4.47·17-s − 2.47·19-s − 0.277·21-s − 2·23-s + 25-s − 1.64·27-s − 0.277·29-s + 10.1·31-s + 1.11·33-s − 35-s + 0.833·37-s − 0.277·39-s − 0.478·41-s + 4.55·43-s − 2.92·45-s − 12.4·47-s + 49-s + 1.24·51-s + 8.40·53-s + 4·55-s + ⋯ |
L(s) = 1 | + 0.160·3-s + 0.447·5-s − 0.377·7-s − 0.974·9-s + 1.20·11-s − 0.277·13-s + 0.0717·15-s + 1.08·17-s − 0.568·19-s − 0.0606·21-s − 0.417·23-s + 0.200·25-s − 0.316·27-s − 0.0515·29-s + 1.81·31-s + 0.193·33-s − 0.169·35-s + 0.136·37-s − 0.0444·39-s − 0.0747·41-s + 0.694·43-s − 0.435·45-s − 1.80·47-s + 0.142·49-s + 0.174·51-s + 1.15·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.041949213\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.041949213\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 0.277T + 3T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 0.277T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 0.833T + 37T^{2} \) |
| 41 | \( 1 + 0.478T + 41T^{2} \) |
| 43 | \( 1 - 4.55T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 8.40T + 53T^{2} \) |
| 59 | \( 1 - 8.27T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 7.92T + 67T^{2} \) |
| 71 | \( 1 - 6.95T + 71T^{2} \) |
| 73 | \( 1 - 9.29T + 73T^{2} \) |
| 79 | \( 1 - 3.92T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 5.44T + 97T^{2} \) |
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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
−8.497868896644744069790205872847, −7.996249731522876220331094466434, −6.90323357034896553133351676906, −6.28076301444336151268748374027, −5.70223646251819128416681119432, −4.75402631086608546789204319248, −3.77804985304861626883838381751, −3.00689042481915673785306741171, −2.08081036016496260973818813323, −0.836237866486909630698825403306,
0.836237866486909630698825403306, 2.08081036016496260973818813323, 3.00689042481915673785306741171, 3.77804985304861626883838381751, 4.75402631086608546789204319248, 5.70223646251819128416681119432, 6.28076301444336151268748374027, 6.90323357034896553133351676906, 7.996249731522876220331094466434, 8.497868896644744069790205872847