L(s) = 1 | + (0.707 + 0.707i)5-s + (−1 + i)13-s + (1.41 − 1.41i)17-s + 1.00i·25-s + 1.41·29-s + (1 + i)37-s + 1.41i·41-s + i·49-s + (−1.41 − 1.41i)53-s − 1.41·65-s + (−1 + i)73-s + 2.00·85-s + 1.41·89-s + (−1 − i)97-s − 1.41i·101-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)5-s + (−1 + i)13-s + (1.41 − 1.41i)17-s + 1.00i·25-s + 1.41·29-s + (1 + i)37-s + 1.41i·41-s + i·49-s + (−1.41 − 1.41i)53-s − 1.41·65-s + (−1 + i)73-s + 2.00·85-s + 1.41·89-s + (−1 − i)97-s − 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.403671547\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.403671547\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1 - i)T - iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.350671075800394598446294603861, −8.186695361437362424015718446474, −7.43822296514295709863452987168, −6.76622273291187949016051496385, −6.11996903532565470714220458433, −5.11986727781763094305097409929, −4.52546195238292836567902063255, −3.11611641826183295277046956250, −2.63691476428635955911080118077, −1.38687071334865829826438095344,
1.00487331015154251525109507367, 2.15744213240418502592389045467, 3.15546935633479616241628653306, 4.22920663736593155309763152055, 5.14497657420040012058525392844, 5.71841996654793039583486992997, 6.41054305811444356417102486091, 7.60403561805083509603135668208, 8.051053988755101448938724834908, 8.890771953579775785733471181651