| L(s) = 1 | + 0.0947·3-s − 5-s + 0.826·7-s − 2.99·9-s + 2.99·13-s − 0.0947·15-s − 0.662·19-s + 0.0783·21-s + 0.473·23-s + 25-s − 0.567·27-s − 4·29-s − 4.64·31-s − 0.826·35-s + 5.98·37-s + 0.283·39-s + 0.394·41-s + 6.96·43-s + 2.99·45-s − 3.88·47-s − 6.31·49-s − 0.801·53-s − 0.0628·57-s − 1.71·59-s − 2.58·61-s − 2.47·63-s − 2.99·65-s + ⋯ |
| L(s) = 1 | + 0.0547·3-s − 0.447·5-s + 0.312·7-s − 0.997·9-s + 0.829·13-s − 0.0244·15-s − 0.152·19-s + 0.0171·21-s + 0.0986·23-s + 0.200·25-s − 0.109·27-s − 0.742·29-s − 0.834·31-s − 0.139·35-s + 0.983·37-s + 0.0453·39-s + 0.0616·41-s + 1.06·43-s + 0.445·45-s − 0.567·47-s − 0.902·49-s − 0.110·53-s − 0.00831·57-s − 0.223·59-s − 0.330·61-s − 0.311·63-s − 0.370·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - 0.0947T + 3T^{2} \) |
| 7 | \( 1 - 0.826T + 7T^{2} \) |
| 13 | \( 1 - 2.99T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 0.662T + 19T^{2} \) |
| 23 | \( 1 - 0.473T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 - 5.98T + 37T^{2} \) |
| 41 | \( 1 - 0.394T + 41T^{2} \) |
| 43 | \( 1 - 6.96T + 43T^{2} \) |
| 47 | \( 1 + 3.88T + 47T^{2} \) |
| 53 | \( 1 + 0.801T + 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 + 2.58T + 61T^{2} \) |
| 67 | \( 1 - 5.69T + 67T^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 5.90T + 83T^{2} \) |
| 89 | \( 1 - 4.32T + 89T^{2} \) |
| 97 | \( 1 - 6.50T + 97T^{2} \) |
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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
−7.905557660898495474779193856148, −7.37811848534411513891527713294, −6.33438144126661484134299403308, −5.80049086187115752496346963005, −4.97813194788055542694192850371, −4.08199470780502729531894387540, −3.36043435530611953725181416520, −2.48555648855299335804546112440, −1.34166816743814632468553754744, 0,
1.34166816743814632468553754744, 2.48555648855299335804546112440, 3.36043435530611953725181416520, 4.08199470780502729531894387540, 4.97813194788055542694192850371, 5.80049086187115752496346963005, 6.33438144126661484134299403308, 7.37811848534411513891527713294, 7.905557660898495474779193856148
