L(s) = 1 | + 0.517i·2-s + (−0.608 + 0.793i)3-s + 0.732·4-s + (−0.410 − 0.315i)6-s + 0.896i·8-s + (−0.258 − 0.965i)9-s + (−0.445 + 0.580i)12-s + 0.261i·13-s + 0.267·16-s + (0.499 − 0.133i)18-s − i·23-s + (−0.711 − 0.545i)24-s + 25-s − 0.135·26-s + (0.923 + 0.382i)27-s + ⋯ |
L(s) = 1 | + 0.517i·2-s + (−0.608 + 0.793i)3-s + 0.732·4-s + (−0.410 − 0.315i)6-s + 0.896i·8-s + (−0.258 − 0.965i)9-s + (−0.445 + 0.580i)12-s + 0.261i·13-s + 0.267·16-s + (0.499 − 0.133i)18-s − i·23-s + (−0.711 − 0.545i)24-s + 25-s − 0.135·26-s + (0.923 + 0.382i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.230163172\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230163172\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.608 - 0.793i)T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 2 | \( 1 - 0.517iT - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 0.261iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 - 1.98iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.261T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.58T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + 1.21iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.784662800760318743432204517755, −8.514802709911770103951391926470, −7.25410567769072860055569874922, −6.72047933601997219166864040845, −6.16837939610578988903606094946, −5.11930242088745407209621687699, −4.83629761710723857283495595502, −3.53499511585774534729600922854, −2.80878528002281230797155416319, −1.41564203404237005196404403829,
0.822766434379721603333092941062, 1.93720979162199527097191805763, 2.64555491591659331935201148681, 3.69044964697093116059559697178, 4.75791341722479969192516981708, 5.80911595638999091095668446108, 6.22729950196543297586732963650, 7.09000122051622853126416159892, 7.67930156968148501547320148644, 8.303714618774464678427308332915