| 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
−7.74466924239375226317299687955, −6.85958178635551033340510103032, −6.53952869034718401389527761681, −5.70830122294491738999958327143, −4.80898281286861673458744653151, −4.33008232893092221770613993930, −3.02915785969883678441164815471, −2.56978697251351836306978245355, −1.52245246961098847665838456627, −0.46719554701026393159259236004,
0.943014542728720522841855419825, 2.06102239810051673830391937656, 3.07216068103367636387873485654, 3.79860519254192723015592413874, 4.41650383617104617423622218576, 5.23969616779960001421657174343, 6.00744439580916140424832084678, 6.48223936734841780920709762603, 7.58030890858699567402162748103, 7.998671026220578444218587192321
