L(s) = 1 | + (−1.11 + 0.866i)2-s + (0.500 − 1.93i)4-s + 3.87i·7-s + (1.11 + 2.59i)8-s + 4·11-s + 2.23·13-s + (−3.35 − 4.33i)14-s + (−3.5 − 1.93i)16-s + 17-s + (4 + 1.73i)19-s + (−4.47 + 3.46i)22-s − 3.87i·23-s + 5·25-s + (−2.50 + 1.93i)26-s + (7.50 + 1.93i)28-s − 2.23·29-s + ⋯ |
L(s) = 1 | + (−0.790 + 0.612i)2-s + (0.250 − 0.968i)4-s + 1.46i·7-s + (0.395 + 0.918i)8-s + 1.20·11-s + 0.620·13-s + (−0.896 − 1.15i)14-s + (−0.875 − 0.484i)16-s + 0.242·17-s + (0.917 + 0.397i)19-s + (−0.953 + 0.738i)22-s − 0.807i·23-s + 25-s + (−0.490 + 0.379i)26-s + (1.41 + 0.365i)28-s − 0.415·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00226 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00226 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.239090386\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239090386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.11 - 0.866i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 3.87iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 23 | \( 1 + 3.87iT - 23T^{2} \) |
| 29 | \( 1 + 2.23T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 7.74iT - 47T^{2} \) |
| 53 | \( 1 - 6.70T + 53T^{2} \) |
| 59 | \( 1 - 8.66iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 13.8iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−9.353077354477782944097231550348, −9.019985407699942855136068984550, −8.409985537434888874205223597233, −7.36182063328726693664030371700, −6.55030038313067356791522061656, −5.76698663820207423569651038928, −5.16626880559771840947659157963, −3.72861840734921393184791974568, −2.37835017173476775461880874195, −1.23811663761057897165943056332,
0.801612224212042745221928357707, 1.69169423823432059749901739149, 3.48799898464237846960042740784, 3.71953514980628509405165988841, 4.99224023954627768010738105219, 6.47750978757067981337975666506, 7.14789241258089456609487334267, 7.74398380302347205834247896771, 8.812075374145664342553264918769, 9.420489885194927901830215588032