L(s) = 1 | + (0.866 + 0.5i)3-s + (1.61 + 2.79i)5-s + (−1.82 + 1.91i)7-s + (0.499 + 0.866i)9-s + (1.10 − 1.91i)11-s + 5.08·13-s + 3.22i·15-s + (−2.73 − 1.57i)17-s + (−2.93 + 1.69i)19-s + (−2.53 + 0.743i)21-s + (−2.65 + 1.53i)23-s + (−2.70 + 4.69i)25-s + 0.999i·27-s + 9.88i·29-s + (1.01 − 1.75i)31-s + ⋯ |
L(s) = 1 | + (0.499 + 0.288i)3-s + (0.721 + 1.25i)5-s + (−0.690 + 0.723i)7-s + (0.166 + 0.288i)9-s + (0.333 − 0.577i)11-s + 1.40·13-s + 0.833i·15-s + (−0.663 − 0.383i)17-s + (−0.673 + 0.388i)19-s + (−0.554 + 0.162i)21-s + (−0.553 + 0.319i)23-s + (−0.541 + 0.938i)25-s + 0.192i·27-s + 1.83i·29-s + (0.182 − 0.315i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0638 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0638 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36205 + 1.27764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36205 + 1.27764i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (1.82 - 1.91i)T \) |
good | 5 | \( 1 + (-1.61 - 2.79i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.10 + 1.91i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.08T + 13T^{2} \) |
| 17 | \( 1 + (2.73 + 1.57i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.93 - 1.69i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.65 - 1.53i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9.88iT - 29T^{2} \) |
| 31 | \( 1 + (-1.01 + 1.75i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.798 - 0.460i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.96iT - 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 + (1.06 + 1.83i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.12 + 1.80i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.6 - 6.14i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.34 + 10.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.40 + 7.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.7iT - 71T^{2} \) |
| 73 | \( 1 + (-7.82 - 4.51i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.60 + 5.54i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.57iT - 83T^{2} \) |
| 89 | \( 1 + (6.32 - 3.65i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.2iT - 97T^{2} \) |
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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
−10.73582125514379427646189643351, −9.823153973753315537385095890067, −9.015986607257815792241795531547, −8.372023903403442056928612326288, −6.93930840294689082681796984278, −6.31298204975233665502207080516, −5.54884164035556428404032881550, −3.82264304419658413021463247157, −3.07873524296110395642632978956, −2.01875976711825331329085039220,
0.995733397619025911809698202723, 2.20973577675256316594294482250, 3.85365478627166377486861513322, 4.55446054114757420353615936967, 6.04770769376043730106223130345, 6.55228836437397327591937323468, 7.86502205040004656077272121393, 8.675708441449583180806885017476, 9.339977377136437396499423999982, 10.07594355054260957365116158620