L(s) = 1 | + (1.22 − 1.22i)3-s − 2.44·7-s − 2.99i·9-s + 4.89·11-s + 2i·13-s + 6·17-s − 4.89i·19-s + (−2.99 + 2.99i)21-s + 2.44i·23-s + (−3.67 − 3.67i)27-s − 9.79i·31-s + (5.99 − 5.99i)33-s + 2i·37-s + (2.44 + 2.44i)39-s − 6i·41-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s − 0.925·7-s − 0.999i·9-s + 1.47·11-s + 0.554i·13-s + 1.45·17-s − 1.12i·19-s + (−0.654 + 0.654i)21-s + 0.510i·23-s + (−0.707 − 0.707i)27-s − 1.75i·31-s + (1.04 − 1.04i)33-s + 0.328i·37-s + (0.392 + 0.392i)39-s − 0.937i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.040068301\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.040068301\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 2.44iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.79iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + 12.2iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 7.34T + 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 4.89iT - 79T^{2} \) |
| 83 | \( 1 + 7.34iT - 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.466349627227231564352294825339, −8.895167607450200966128226608442, −7.924474869805007932499353997698, −6.96286599494741860273561082632, −6.56373813522172671280857867397, −5.55090204027147250939065129680, −3.99150396075454606348095535751, −3.39812385153242594565528537706, −2.20013094030594625422467046276, −0.902389848616965654331420979902,
1.45371600830483743795150592584, 3.12310875135830545435174305429, 3.52101798879906307408989502852, 4.59038584245134141689728967717, 5.72130294843633234431926905971, 6.54238952540630686659318318380, 7.60016923159732856827588423333, 8.406670126359618053316683473939, 9.200960922542492783039898380164, 9.908020907559774049520059444414