L(s) = 1 | + i·3-s − 0.747i·7-s − 9-s + 0.209·11-s − 2.50i·13-s + 6.81i·17-s − 1.24·19-s + 0.747·21-s + 3.25i·23-s − i·27-s + 5.18·29-s + 3.73·31-s + 0.209i·33-s − 1.96i·37-s + 2.50·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.282i·7-s − 0.333·9-s + 0.0630·11-s − 0.694i·13-s + 1.65i·17-s − 0.285·19-s + 0.163·21-s + 0.677i·23-s − 0.192i·27-s + 0.962·29-s + 0.671·31-s + 0.0363i·33-s − 0.323i·37-s + 0.400·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.650363540\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.650363540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.747iT - 7T^{2} \) |
| 11 | \( 1 - 0.209T + 11T^{2} \) |
| 13 | \( 1 + 2.50iT - 13T^{2} \) |
| 17 | \( 1 - 6.81iT - 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 3.25iT - 23T^{2} \) |
| 29 | \( 1 - 5.18T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 + 1.96iT - 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 + 12.7iT - 43T^{2} \) |
| 47 | \( 1 - 6.48iT - 47T^{2} \) |
| 53 | \( 1 - 4.14iT - 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 2.91iT - 67T^{2} \) |
| 71 | \( 1 - 6.55T + 71T^{2} \) |
| 73 | \( 1 + 2.51iT - 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 5.67iT - 83T^{2} \) |
| 89 | \( 1 - 0.921T + 89T^{2} \) |
| 97 | \( 1 - 15.5iT - 97T^{2} \) |
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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
−7.998671026220578444218587192321, −7.58030890858699567402162748103, −6.48223936734841780920709762603, −6.00744439580916140424832084678, −5.23969616779960001421657174343, −4.41650383617104617423622218576, −3.79860519254192723015592413874, −3.07216068103367636387873485654, −2.06102239810051673830391937656, −0.943014542728720522841855419825,
0.46719554701026393159259236004, 1.52245246961098847665838456627, 2.56978697251351836306978245355, 3.02915785969883678441164815471, 4.33008232893092221770613993930, 4.80898281286861673458744653151, 5.70830122294491738999958327143, 6.53952869034718401389527761681, 6.85958178635551033340510103032, 7.74466924239375226317299687955