| L(s) = 1 | + (−2.63 + 0.197i)2-s + (1.19 + 1.25i)3-s + (4.91 − 0.741i)4-s + (−0.535 − 2.34i)5-s + (−3.38 − 3.07i)6-s + (2.62 + 0.334i)7-s + (−7.65 + 1.74i)8-s + (−0.156 + 2.99i)9-s + (1.87 + 6.07i)10-s + (−1.46 + 3.03i)11-s + (6.79 + 5.29i)12-s + (−3.74 + 0.280i)13-s + (−6.97 − 0.362i)14-s + (2.31 − 3.47i)15-s + (10.2 − 3.17i)16-s + (7.66 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (−1.86 + 0.139i)2-s + (0.688 + 0.725i)3-s + (2.45 − 0.370i)4-s + (−0.239 − 1.04i)5-s + (−1.38 − 1.25i)6-s + (0.991 + 0.126i)7-s + (−2.70 + 0.617i)8-s + (−0.0521 + 0.998i)9-s + (0.592 + 1.92i)10-s + (−0.441 + 0.915i)11-s + (1.96 + 1.52i)12-s + (−1.03 + 0.0777i)13-s + (−1.86 − 0.0968i)14-s + (0.596 − 0.896i)15-s + (2.57 − 0.794i)16-s + (1.85 + 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.709025 + 0.355899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.709025 + 0.355899i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.19 - 1.25i)T \) |
| 7 | \( 1 + (-2.62 - 0.334i)T \) |
| good | 2 | \( 1 + (2.63 - 0.197i)T + (1.97 - 0.298i)T^{2} \) |
| 5 | \( 1 + (0.535 + 2.34i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (1.46 - 3.03i)T + (-6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (3.74 - 0.280i)T + (12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-7.66 - 1.15i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.55 - 0.900i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.221 - 0.176i)T + (5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (-4.27 + 1.67i)T + (21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (0.817 + 0.471i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.22 - 3.11i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.85 + 2.65i)T + (3.06 - 40.8i)T^{2} \) |
| 43 | \( 1 + (-3.13 - 2.90i)T + (3.21 + 42.8i)T^{2} \) |
| 47 | \( 1 + (-0.727 - 9.71i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (-12.6 - 4.96i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (9.46 + 8.78i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-2.18 + 14.5i)T + (-58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (4.15 - 7.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.47 - 4.36i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (3.00 - 4.40i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (1.83 + 3.18i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.0613 - 0.818i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (0.425 - 5.68i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (-1.59 - 0.918i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.78655770082606658542687206725, −9.948880442620650420862050601225, −9.494413913063870477633717431946, −8.528347883623208909800201057812, −7.85946173525495835778051924375, −7.46350872202162715893688133716, −5.51444515324251130245807533060, −4.55221439268368439324648084479, −2.63414060929052463277495876380, −1.37138610543068917422332501855,
0.969993924322073100013935316761, 2.44895957026959814764797747398, 3.22117762821061214925242965478, 5.74403984182237627316233624612, 7.16781713136494025677853542865, 7.44322288810613876191085327752, 8.190346860944224124789765496165, 9.009980010567988868154161159689, 10.12602208045357384029238230309, 10.65956283457691386075884851840
