| L(s) = 1 | − 3-s + 11-s + 17-s + 19-s + 27-s − 33-s − 41-s + 2·43-s + 49-s − 51-s − 57-s − 2·59-s − 67-s + 73-s − 81-s − 83-s − 89-s − 2·97-s − 107-s + 113-s + ⋯ |
| L(s) = 1 | − 3-s + 11-s + 17-s + 19-s + 27-s − 33-s − 41-s + 2·43-s + 49-s − 51-s − 57-s − 2·59-s − 67-s + 73-s − 81-s − 83-s − 89-s − 2·97-s − 107-s + 113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7444789510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7444789510\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−10.61063892686163918452289121290, −9.684548852188609098458975323436, −8.912860114274773222361978368772, −7.78826680907517205754196796507, −6.89218535721784810346737333139, −5.96433838249342217139008185107, −5.34392177496741387155959719539, −4.21356259088563849915972179100, −3.03264549984213012216405315951, −1.21475066340725495173025466188,
1.21475066340725495173025466188, 3.03264549984213012216405315951, 4.21356259088563849915972179100, 5.34392177496741387155959719539, 5.96433838249342217139008185107, 6.89218535721784810346737333139, 7.78826680907517205754196796507, 8.912860114274773222361978368772, 9.684548852188609098458975323436, 10.61063892686163918452289121290
