L(s) = 1 | + (1.5 + 0.866i)3-s + (−2.5 − 0.866i)7-s + (1.5 + 2.59i)9-s − 5.19i·13-s + (−7.5 + 4.33i)19-s + (−3 − 3.46i)21-s + (2.5 − 4.33i)25-s + 5.19i·27-s + (−1.5 − 0.866i)31-s + (5.5 + 9.52i)37-s + (4.5 − 7.79i)39-s + 13·43-s + (5.5 + 4.33i)49-s − 15·57-s + (−6 + 3.46i)61-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + (−0.944 − 0.327i)7-s + (0.5 + 0.866i)9-s − 1.44i·13-s + (−1.72 + 0.993i)19-s + (−0.654 − 0.755i)21-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (−0.269 − 0.155i)31-s + (0.904 + 1.56i)37-s + (0.720 − 1.24i)39-s + 1.98·43-s + (0.785 + 0.618i)49-s − 1.98·57-s + (−0.768 + 0.443i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11353 + 0.177917i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11353 + 0.177917i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (7.5 - 4.33i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 13T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (1.5 + 0.866i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−14.45671112541079552165042467063, −13.22231233716616750026509569343, −12.57254959801065830060291352320, −10.61244153110935554435975147557, −10.05107202426334850739651401692, −8.733724148116538139149901003415, −7.70081156292783400695111500847, −6.10574114292596037424511130778, −4.25090263617993966958778313709, −2.87999174247487663390578797746,
2.41433171051343936092513682492, 4.08732909027933863056444995936, 6.30791160156048248823933360980, 7.23827633619799554746052242284, 8.863905064716110456257889432496, 9.371346798397679173883712076492, 11.00061518185197420179823481078, 12.43937190505582300927148332302, 13.10525951925068965063688358972, 14.19403850596027687517940994606