L(s) = 1 | − 2.37i·2-s + (−1.69 − 1.69i)3-s − 3.62·4-s + (−0.324 − 0.324i)5-s + (−4.01 + 4.01i)6-s + (1.08 − 1.08i)7-s + 3.84i·8-s + 2.73i·9-s + (−0.768 + 0.768i)10-s + (−0.707 + 0.707i)11-s + (6.12 + 6.12i)12-s + 1.06·13-s + (−2.57 − 2.57i)14-s + 1.09i·15-s + 1.86·16-s + (1.53 + 3.82i)17-s + ⋯ |
L(s) = 1 | − 1.67i·2-s + (−0.977 − 0.977i)3-s − 1.81·4-s + (−0.144 − 0.144i)5-s + (−1.63 + 1.63i)6-s + (0.410 − 0.410i)7-s + 1.35i·8-s + 0.910i·9-s + (−0.242 + 0.242i)10-s + (−0.213 + 0.213i)11-s + (1.76 + 1.76i)12-s + 0.295·13-s + (−0.688 − 0.688i)14-s + 0.283i·15-s + 0.466·16-s + (0.373 + 0.927i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.337975 + 0.573164i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.337975 + 0.573164i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 + (-1.53 - 3.82i)T \) |
good | 2 | \( 1 + 2.37iT - 2T^{2} \) |
| 3 | \( 1 + (1.69 + 1.69i)T + 3iT^{2} \) |
| 5 | \( 1 + (0.324 + 0.324i)T + 5iT^{2} \) |
| 7 | \( 1 + (-1.08 + 1.08i)T - 7iT^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 19 | \( 1 + 6.52iT - 19T^{2} \) |
| 23 | \( 1 + (0.493 - 0.493i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.169 - 0.169i)T + 29iT^{2} \) |
| 31 | \( 1 + (5.69 + 5.69i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.66 + 1.66i)T + 37iT^{2} \) |
| 41 | \( 1 + (2.72 - 2.72i)T - 41iT^{2} \) |
| 43 | \( 1 + 3.22iT - 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 1.24iT - 53T^{2} \) |
| 59 | \( 1 + 10.1iT - 59T^{2} \) |
| 61 | \( 1 + (-9.61 + 9.61i)T - 61iT^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + (-2.38 - 2.38i)T + 71iT^{2} \) |
| 73 | \( 1 + (0.254 + 0.254i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.436 - 0.436i)T - 79iT^{2} \) |
| 83 | \( 1 + 11.0iT - 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + (-4.11 - 4.11i)T + 97iT^{2} \) |
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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
−11.84529688493942471899955573661, −11.13569040222888435174362220396, −10.49706453195737788804985988948, −9.222888043957124286595645120888, −7.914914903956869284930053655145, −6.64286027197311110002446835472, −5.20267782240866814897089861941, −3.92315245571527064146084746850, −2.09841243625747660695872852319, −0.69833244350599368282186931237,
3.93239568467142251141876138689, 5.31413286764615026659722829548, 5.57831096576664333771928322045, 6.92572249150939594916410507301, 8.023736086039533940890803510329, 9.061775574896507224764046582495, 10.15421284235317201246722315459, 11.19857264583823755517255937847, 12.18972365132806073604874773793, 13.64599841095950856939279331788