L(s) = 1 | + i·2-s − 4-s + (−2 + i)5-s − 2i·7-s − i·8-s + (−1 − 2i)10-s + 6·11-s − 2i·13-s + 2·14-s + 16-s + 6i·17-s − 8·19-s + (2 − i)20-s + 6i·22-s − 4i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.755i·7-s − 0.353i·8-s + (−0.316 − 0.632i)10-s + 1.80·11-s − 0.554i·13-s + 0.534·14-s + 0.250·16-s + 1.45i·17-s − 1.83·19-s + (0.447 − 0.223i)20-s + 1.27i·22-s − 0.834i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7774375123\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7774375123\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−8.541929243503319816189804479996, −7.896083085709122792290005318772, −7.05586869779499667883133627318, −6.49580116874956016634666786876, −5.93418255541737317983063657958, −4.37751231813317076756371808500, −4.13988336851074189264126987040, −3.35605328859034433015029230015, −1.71733769180083917199058979112, −0.27882036845845731905840114920,
1.19153188991427960478068159100, 2.21803526142860722166640311296, 3.38857834604733503745634490375, 4.14672469426042080520212108669, 4.72863069580394294238342544199, 5.79309548456594990933325815517, 6.70898617062002831849036029098, 7.46742238702381036867177553708, 8.507011045644721387098044728637, 9.174385658886192060738267710896