L(s) = 1 | + 3·7-s + 3·11-s − 13-s − 5·17-s − 6·19-s − 3·23-s + 2·29-s + 6·31-s + 3·37-s − 5·41-s + 2·49-s + 7·53-s + 61-s + 15·71-s − 2·73-s + 9·77-s − 15·79-s − 6·83-s − 11·89-s − 3·91-s + 97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 15·119-s + ⋯ |
L(s) = 1 | + 1.13·7-s + 0.904·11-s − 0.277·13-s − 1.21·17-s − 1.37·19-s − 0.625·23-s + 0.371·29-s + 1.07·31-s + 0.493·37-s − 0.780·41-s + 2/7·49-s + 0.961·53-s + 0.128·61-s + 1.78·71-s − 0.234·73-s + 1.02·77-s − 1.68·79-s − 0.658·83-s − 1.16·89-s − 0.314·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.37·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.08032971479702, −13.74565510992385, −12.97990399467084, −12.70681078890992, −11.93419807634167, −11.56056680186980, −11.30759650630279, −10.56830429004539, −10.24174363013952, −9.603320958428970, −8.935462541574603, −8.541732680280311, −8.195195531112347, −7.595454235811041, −6.796584242321699, −6.587235435656688, −5.961668703350568, −5.262583207629349, −4.580831889754242, −4.318870472060771, −3.806517117118158, −2.823086479749090, −2.195848567750499, −1.735786497209331, −0.9568078891721680, 0,
0.9568078891721680, 1.735786497209331, 2.195848567750499, 2.823086479749090, 3.806517117118158, 4.318870472060771, 4.580831889754242, 5.262583207629349, 5.961668703350568, 6.587235435656688, 6.796584242321699, 7.595454235811041, 8.195195531112347, 8.541732680280311, 8.935462541574603, 9.603320958428970, 10.24174363013952, 10.56830429004539, 11.30759650630279, 11.56056680186980, 11.93419807634167, 12.70681078890992, 12.97990399467084, 13.74565510992385, 14.08032971479702