L(s) = 1 | − 2.82·5-s + 7-s − 2·11-s + 4.82·13-s − 4·17-s + 19-s + 0.828·23-s + 3.00·25-s + 7.65·29-s + 1.17·31-s − 2.82·35-s − 8.82·37-s − 2·41-s + 1.65·43-s − 10.4·47-s + 49-s − 6·53-s + 5.65·55-s − 4·59-s − 6·61-s − 13.6·65-s + 11.3·67-s − 13.6·71-s + 10·73-s − 2·77-s + 14.8·79-s + 15.6·83-s + ⋯ |
L(s) = 1 | − 1.26·5-s + 0.377·7-s − 0.603·11-s + 1.33·13-s − 0.970·17-s + 0.229·19-s + 0.172·23-s + 0.600·25-s + 1.42·29-s + 0.210·31-s − 0.478·35-s − 1.45·37-s − 0.312·41-s + 0.252·43-s − 1.52·47-s + 0.142·49-s − 0.824·53-s + 0.762·55-s − 0.520·59-s − 0.768·61-s − 1.69·65-s + 1.38·67-s − 1.62·71-s + 1.17·73-s − 0.227·77-s + 1.66·79-s + 1.71·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 8.82T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.48935662020286926813297623600, −6.63994601206952979782139814488, −6.17452172159072746067291518206, −4.97886634915747501555036272943, −4.69512817192142540259275188361, −3.68809063160133001641568423848, −3.28973185598106911419952532302, −2.17917649493817097690997598026, −1.10540297389681392526699458541, 0,
1.10540297389681392526699458541, 2.17917649493817097690997598026, 3.28973185598106911419952532302, 3.68809063160133001641568423848, 4.69512817192142540259275188361, 4.97886634915747501555036272943, 6.17452172159072746067291518206, 6.63994601206952979782139814488, 7.48935662020286926813297623600