L(s) = 1 | + 5-s − 7-s + 2·11-s + 4·13-s + 2·17-s + 6·19-s + 4·23-s + 25-s − 10·29-s + 2·31-s − 35-s + 2·37-s + 10·41-s − 4·43-s + 8·47-s + 49-s + 4·53-s + 2·55-s + 4·59-s + 2·61-s + 4·65-s − 4·67-s − 6·71-s − 2·77-s − 4·79-s − 4·83-s + 2·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.603·11-s + 1.10·13-s + 0.485·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s − 1.85·29-s + 0.359·31-s − 0.169·35-s + 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.549·53-s + 0.269·55-s + 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s − 0.712·71-s − 0.227·77-s − 0.450·79-s − 0.439·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.109665976\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.109665976\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.68457096839619, −15.12423914750925, −14.45695633965625, −14.07117613032547, −13.33873326443047, −13.11571336453219, −12.42864108839531, −11.67319371063765, −11.33738167280086, −10.68408150926627, −10.03964956543165, −9.415448934984064, −9.076034891548603, −8.476909541294977, −7.498306618765079, −7.270069069962307, −6.379448622917594, −5.806632057581882, −5.445534885643446, −4.496332576165618, −3.708049834204960, −3.260933508473273, −2.389148968067256, −1.398213661279340, −0.8103577327146951,
0.8103577327146951, 1.398213661279340, 2.389148968067256, 3.260933508473273, 3.708049834204960, 4.496332576165618, 5.445534885643446, 5.806632057581882, 6.379448622917594, 7.270069069962307, 7.498306618765079, 8.476909541294977, 9.076034891548603, 9.415448934984064, 10.03964956543165, 10.68408150926627, 11.33738167280086, 11.67319371063765, 12.42864108839531, 13.11571336453219, 13.33873326443047, 14.07117613032547, 14.45695633965625, 15.12423914750925, 15.68457096839619