L(s) = 1 | + (−1.36 − 1.36i)2-s + (0.866 + 0.5i)3-s + 2.73i·4-s + (−0.499 − 1.86i)6-s + (2.36 − 2.36i)8-s + (0.499 + 0.866i)9-s + (−1.36 + 2.36i)12-s + i·13-s − 3.73·16-s + (0.5 − 1.86i)18-s − i·23-s + (3.23 − 0.866i)24-s + 25-s + (1.36 − 1.36i)26-s + 0.999i·27-s + ⋯ |
L(s) = 1 | + (−1.36 − 1.36i)2-s + (0.866 + 0.5i)3-s + 2.73i·4-s + (−0.499 − 1.86i)6-s + (2.36 − 2.36i)8-s + (0.499 + 0.866i)9-s + (−1.36 + 2.36i)12-s + i·13-s − 3.73·16-s + (0.5 − 1.86i)18-s − i·23-s + (3.23 − 0.866i)24-s + 25-s + (1.36 − 1.36i)26-s + 0.999i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7869477382\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7869477382\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 + (-0.866 + 0.5i)T \) |
good | 2 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 31 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 2iT - T^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.364731287060467008354394539108, −8.725705416815685779339773465196, −8.282348751421814410122542859343, −7.39661435503404240555190115090, −6.62678357677016197280100112272, −4.71053927755874912365606573989, −4.08342838137541537382530050618, −3.06314956301727082962698382275, −2.41534826216741107223804090846, −1.37099445475855887632262427722,
0.915726320461567854752598713572, 2.05922790361499140130650471231, 3.39657362086873662715205391487, 4.93061985556174955312500072481, 5.69867129524085286195481193168, 6.70254386874767697018534394739, 7.10412934105257900290598490143, 8.033831685052919770827941175704, 8.326749155444075907232842323589, 9.131549916224690491095738961234