L(s) = 1 | + (1.78 − 1.03i)2-s + (−0.627 + 1.61i)3-s + (1.12 − 1.95i)4-s + (0.5 + 0.866i)5-s + (0.543 + 3.53i)6-s + (−0.00953 − 2.64i)7-s − 0.527i·8-s + (−2.21 − 2.02i)9-s + (1.78 + 1.03i)10-s + (−4.06 − 2.34i)11-s + (2.44 + 3.04i)12-s + 0.638i·13-s + (−2.74 − 4.71i)14-s + (−1.71 + 0.263i)15-s + (1.71 + 2.96i)16-s + (−2.07 + 3.59i)17-s + ⋯ |
L(s) = 1 | + (1.26 − 0.729i)2-s + (−0.362 + 0.932i)3-s + (0.563 − 0.976i)4-s + (0.223 + 0.387i)5-s + (0.221 + 1.44i)6-s + (−0.00360 − 0.999i)7-s − 0.186i·8-s + (−0.737 − 0.675i)9-s + (0.564 + 0.326i)10-s + (−1.22 − 0.707i)11-s + (0.705 + 0.879i)12-s + 0.177i·13-s + (−0.733 − 1.26i)14-s + (−0.442 + 0.0680i)15-s + (0.427 + 0.741i)16-s + (−0.503 + 0.871i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60004 - 0.193350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60004 - 0.193350i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.627 - 1.61i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.00953 + 2.64i)T \) |
good | 2 | \( 1 + (-1.78 + 1.03i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (4.06 + 2.34i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.638iT - 13T^{2} \) |
| 17 | \( 1 + (2.07 - 3.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.776 - 0.448i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.89 + 3.40i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.14iT - 29T^{2} \) |
| 31 | \( 1 + (2.02 + 1.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.69 - 9.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4.10T + 41T^{2} \) |
| 43 | \( 1 - 3.14T + 43T^{2} \) |
| 47 | \( 1 + (3.40 + 5.89i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.96 - 1.13i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.254 - 0.440i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.48 + 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.41 - 4.18i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.22iT - 71T^{2} \) |
| 73 | \( 1 + (-12.5 - 7.22i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.54 + 7.86i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.76T + 83T^{2} \) |
| 89 | \( 1 + (6.90 + 11.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.9iT - 97T^{2} \) |
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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
−13.56350154359597975818388352208, −12.91067084024626815012074462262, −11.42565350041195789218220765765, −10.78901529827465079229752880765, −10.12794029184922457217913806866, −8.356402065966805306165534145306, −6.42635432795906055589815040963, −5.19942907610908329166890030836, −4.12623637892311151497127713046, −2.93940084751749054223419473568,
2.61578888808864390979676742143, 4.99580413979799486386904975575, 5.54767995398334892871223679998, 6.82937242029342328101228778521, 7.81518613603387369234564069294, 9.303094200776250935996873343872, 11.10690548318724000135432995110, 12.36163389743398264189885353475, 12.86637800941429432627859360542, 13.59057006785725356013798619623