L(s) = 1 | − 7-s + 11-s − 2·13-s + 4·17-s + 2·19-s − 5·23-s − 29-s + 2·31-s − 3·37-s − 12·41-s + 11·43-s − 2·47-s + 49-s + 6·53-s − 10·59-s + 4·61-s + 67-s − 3·71-s − 77-s + 9·79-s + 2·83-s + 6·89-s + 2·91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 0.301·11-s − 0.554·13-s + 0.970·17-s + 0.458·19-s − 1.04·23-s − 0.185·29-s + 0.359·31-s − 0.493·37-s − 1.87·41-s + 1.67·43-s − 0.291·47-s + 1/7·49-s + 0.824·53-s − 1.30·59-s + 0.512·61-s + 0.122·67-s − 0.356·71-s − 0.113·77-s + 1.01·79-s + 0.219·83-s + 0.635·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.62909409442327, −15.11030645864763, −14.52229293506265, −13.95613631789118, −13.65876263743255, −12.87741459828315, −12.24085494952945, −12.02627466773539, −11.41513653017299, −10.60963920373455, −10.11973416381198, −9.691993342401546, −9.103891276860232, −8.449222188013173, −7.776309234254699, −7.348959257845815, −6.632928621775109, −6.050212368817270, −5.416353062300799, −4.839576204713527, −3.980860459549969, −3.461858747286737, −2.723351202830636, −1.917846329902571, −1.044101483706880, 0,
1.044101483706880, 1.917846329902571, 2.723351202830636, 3.461858747286737, 3.980860459549969, 4.839576204713527, 5.416353062300799, 6.050212368817270, 6.632928621775109, 7.348959257845815, 7.776309234254699, 8.449222188013173, 9.103891276860232, 9.691993342401546, 10.11973416381198, 10.60963920373455, 11.41513653017299, 12.02627466773539, 12.24085494952945, 12.87741459828315, 13.65876263743255, 13.95613631789118, 14.52229293506265, 15.11030645864763, 15.62909409442327