L(s) = 1 | − 3.61·5-s + 7-s + 5.85·11-s + 5.23·13-s + 2.47·17-s − 19-s + 5.70·23-s + 8.09·25-s + 0.854·29-s + 4.47·31-s − 3.61·35-s − 0.0901·37-s + 6.09·41-s + 6.85·43-s + 11.8·47-s + 49-s + 3.38·53-s − 21.1·55-s − 4.09·59-s − 7.85·61-s − 18.9·65-s − 14.9·67-s − 13.5·71-s + 8.76·73-s + 5.85·77-s − 6.85·79-s − 4·83-s + ⋯ |
L(s) = 1 | − 1.61·5-s + 0.377·7-s + 1.76·11-s + 1.45·13-s + 0.599·17-s − 0.229·19-s + 1.19·23-s + 1.61·25-s + 0.158·29-s + 0.803·31-s − 0.611·35-s − 0.0148·37-s + 0.951·41-s + 1.04·43-s + 1.72·47-s + 0.142·49-s + 0.464·53-s − 2.85·55-s − 0.532·59-s − 1.00·61-s − 2.34·65-s − 1.82·67-s − 1.60·71-s + 1.02·73-s + 0.667·77-s − 0.771·79-s − 0.439·83-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
\(L(1)\) |
\(\approx\) |
\(2.242996608\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.242996608\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.61T + 5T^{2} \) |
| 11 | \( 1 - 5.85T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 0.854T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 + 0.0901T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 + 7.85T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 + 3.56T + 97T^{2} \) |
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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
−7.56227729721006237410556702592, −7.21770093575038299974209531980, −6.33167366566478104734830640402, −5.80406933911011178746656381321, −4.56491387738026598295641692599, −4.18608526971003170217114372318, −3.59643036998779116244082795060, −2.87552512628305221414259723137, −1.34482670994166467909836355691, −0.848867351532436814216510324145,
0.848867351532436814216510324145, 1.34482670994166467909836355691, 2.87552512628305221414259723137, 3.59643036998779116244082795060, 4.18608526971003170217114372318, 4.56491387738026598295641692599, 5.80406933911011178746656381321, 6.33167366566478104734830640402, 7.21770093575038299974209531980, 7.56227729721006237410556702592