L(s) = 1 | + (1.89 + 1.18i)5-s − 4.20i·7-s + 11-s − 2.90i·13-s + 2.41i·17-s − 6.16·19-s − 8.05i·23-s + (2.19 + 4.49i)25-s − 5.68·29-s + 1.52·31-s + (4.98 − 7.97i)35-s + 0.577i·37-s − 3.30·41-s − 10.6i·43-s + 0.355i·47-s + ⋯ |
L(s) = 1 | + (0.848 + 0.529i)5-s − 1.58i·7-s + 0.301·11-s − 0.804i·13-s + 0.585i·17-s − 1.41·19-s − 1.67i·23-s + (0.438 + 0.898i)25-s − 1.05·29-s + 0.273·31-s + (0.842 − 1.34i)35-s + 0.0949i·37-s − 0.515·41-s − 1.63i·43-s + 0.0518i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.469369570\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.469369570\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.89 - 1.18i)T \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + 4.20iT - 7T^{2} \) |
| 13 | \( 1 + 2.90iT - 13T^{2} \) |
| 17 | \( 1 - 2.41iT - 17T^{2} \) |
| 19 | \( 1 + 6.16T + 19T^{2} \) |
| 23 | \( 1 + 8.05iT - 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 0.577iT - 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 - 0.355iT - 47T^{2} \) |
| 53 | \( 1 + 9.95iT - 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 8.90T + 61T^{2} \) |
| 67 | \( 1 - 5.40iT - 67T^{2} \) |
| 71 | \( 1 + 6.61T + 71T^{2} \) |
| 73 | \( 1 - 6.21iT - 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 10.7iT - 83T^{2} \) |
| 89 | \( 1 - 7.52T + 89T^{2} \) |
| 97 | \( 1 - 3.33iT - 97T^{2} \) |
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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
−8.269816031898956059808943659922, −7.28731940555434405089688660549, −6.75975244384843622160273116623, −6.16328793220433587584515248713, −5.25980370368917273964491091920, −4.24174833251788626914519701346, −3.68661857612896752373810946901, −2.58335349859505058533711422437, −1.63085609127312872679686930707, −0.38890226380687495389086476135,
1.55183318387669615017164809359, 2.15497442742080613914845297348, 3.08820482582900824926257348965, 4.32277923553295779799954737077, 5.04856200523211060879000255080, 5.87728806621565676626051058196, 6.19060261156531663432504143380, 7.19277899401623199408932252765, 8.171923109680712203941214335270, 8.958981020436322924737302168608