| L(s) = 1 | + 3-s + 1.03·5-s − 2.44·7-s + 9-s + 5.46·11-s − 4.24·13-s + 1.03·15-s + 3.46·17-s + 0.535·19-s − 2.44·21-s + 2.82·23-s − 3.92·25-s + 27-s + 5.93·29-s − 7.34·31-s + 5.46·33-s − 2.53·35-s + 9.14·37-s − 4.24·39-s + 11.4·41-s + 3.46·43-s + 1.03·45-s − 2.82·47-s − 1.00·49-s + 3.46·51-s + 9.52·53-s + 5.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.462·5-s − 0.925·7-s + 0.333·9-s + 1.64·11-s − 1.17·13-s + 0.267·15-s + 0.840·17-s + 0.122·19-s − 0.534·21-s + 0.589·23-s − 0.785·25-s + 0.192·27-s + 1.10·29-s − 1.31·31-s + 0.951·33-s − 0.428·35-s + 1.50·37-s − 0.679·39-s + 1.79·41-s + 0.528·43-s + 0.154·45-s − 0.412·47-s − 0.142·49-s + 0.485·51-s + 1.30·53-s + 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.485416524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.485416524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 1.03T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 0.535T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 9.52T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 + 1.46T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−9.008800720886589346775099659885, −7.84480845780664712473014919068, −7.26145168689405851910480458934, −6.42214236224308750682934356422, −5.86419575817852579177074794099, −4.72462497404040423643125040893, −3.86221845182884493905404584267, −3.07709379239301356467124176969, −2.16964616267818509208410897429, −0.963120409678928536648681214448,
0.963120409678928536648681214448, 2.16964616267818509208410897429, 3.07709379239301356467124176969, 3.86221845182884493905404584267, 4.72462497404040423643125040893, 5.86419575817852579177074794099, 6.42214236224308750682934356422, 7.26145168689405851910480458934, 7.84480845780664712473014919068, 9.008800720886589346775099659885
