| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 4·17-s − 6·19-s + 21-s − 3·23-s − 4·25-s − 27-s − 5·29-s − 4·31-s + 33-s + 35-s + 10·37-s + 39-s − 7·41-s + 5·43-s − 45-s + 8·47-s − 6·49-s − 4·51-s − 2·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.258·15-s + 0.970·17-s − 1.37·19-s + 0.218·21-s − 0.625·23-s − 4/5·25-s − 0.192·27-s − 0.928·29-s − 0.718·31-s + 0.174·33-s + 0.169·35-s + 1.64·37-s + 0.160·39-s − 1.09·41-s + 0.762·43-s − 0.149·45-s + 1.16·47-s − 6/7·49-s − 0.560·51-s − 0.274·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7790776677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7790776677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.922141133282957081347693012438, −7.27040167006729281926148116963, −6.53970236999841359576341200110, −5.78732052149298929395926507576, −5.30623077433658276527267218803, −4.19239954282115829623991362810, −3.84171748440183033544955091841, −2.71239922675723935095022507295, −1.78578956512926085740154221886, −0.45215861285149412929974845039,
0.45215861285149412929974845039, 1.78578956512926085740154221886, 2.71239922675723935095022507295, 3.84171748440183033544955091841, 4.19239954282115829623991362810, 5.30623077433658276527267218803, 5.78732052149298929395926507576, 6.53970236999841359576341200110, 7.27040167006729281926148116963, 7.922141133282957081347693012438
