| L(s) = 1 | − 3.37·3-s − 5-s − 3.37·7-s + 8.37·9-s − 11-s − 2·13-s + 3.37·15-s + 1.37·17-s + 0.627·19-s + 11.3·21-s − 2.74·23-s + 25-s − 18.1·27-s − 1.37·29-s − 3.37·31-s + 3.37·33-s + 3.37·35-s − 9.37·37-s + 6.74·39-s − 11.4·41-s − 4·43-s − 8.37·45-s − 2.74·47-s + 4.37·49-s − 4.62·51-s + 4.11·53-s + 55-s + ⋯ |
| L(s) = 1 | − 1.94·3-s − 0.447·5-s − 1.27·7-s + 2.79·9-s − 0.301·11-s − 0.554·13-s + 0.870·15-s + 0.332·17-s + 0.144·19-s + 2.48·21-s − 0.572·23-s + 0.200·25-s − 3.48·27-s − 0.254·29-s − 0.605·31-s + 0.587·33-s + 0.570·35-s − 1.54·37-s + 1.07·39-s − 1.79·41-s − 0.609·43-s − 1.24·45-s − 0.400·47-s + 0.624·49-s − 0.648·51-s + 0.565·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1839588879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1839588879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 3.37T + 3T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 5.37T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 1.25T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.568295078862601154141834675733, −7.40578008556889700755969915125, −6.97014231300037966282104508460, −6.32481062830683499728239128891, −5.54025206041703473261040644776, −5.02250042297340213860114266036, −4.05185057878905144686531053649, −3.25148845019638194202190276875, −1.68612129142009338092368317542, −0.27529532082538466168583224377,
0.27529532082538466168583224377, 1.68612129142009338092368317542, 3.25148845019638194202190276875, 4.05185057878905144686531053649, 5.02250042297340213860114266036, 5.54025206041703473261040644776, 6.32481062830683499728239128891, 6.97014231300037966282104508460, 7.40578008556889700755969915125, 8.568295078862601154141834675733
