L(s) = 1 | + 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (4.99 − 0.134i)5-s − 2.44·6-s − 11.0·7-s − 2.82i·8-s − 2.99·9-s + (0.190 + 7.06i)10-s − 9.44i·11-s − 3.46i·12-s − 22.3i·13-s − 15.6i·14-s + (0.233 + 8.65i)15-s + 4.00·16-s + 29.3·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (0.999 − 0.0269i)5-s − 0.408·6-s − 1.58·7-s − 0.353i·8-s − 0.333·9-s + (0.0190 + 0.706i)10-s − 0.858i·11-s − 0.288i·12-s − 1.72i·13-s − 1.11i·14-s + (0.0155 + 0.577i)15-s + 0.250·16-s + 1.72·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.665171368\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.665171368\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + (-4.99 + 0.134i)T \) |
| 23 | \( 1 + (-20.6 + 10.0i)T \) |
good | 7 | \( 1 + 11.0T + 49T^{2} \) |
| 11 | \( 1 + 9.44iT - 121T^{2} \) |
| 13 | \( 1 + 22.3iT - 169T^{2} \) |
| 17 | \( 1 - 29.3T + 289T^{2} \) |
| 19 | \( 1 - 31.8iT - 361T^{2} \) |
| 29 | \( 1 - 21.8T + 841T^{2} \) |
| 31 | \( 1 + 33.9T + 961T^{2} \) |
| 37 | \( 1 - 7.59T + 1.36e3T^{2} \) |
| 41 | \( 1 - 55.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 22.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 77.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 39.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 70.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 31.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 21.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 31.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 64.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 8.93iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.63T + 6.88e3T^{2} \) |
| 89 | \( 1 - 153. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 71.0T + 9.40e3T^{2} \) |
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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
−10.09524398058894532092843086551, −9.641664035208406958689214288758, −8.641613966840347910869895773254, −7.77731542656729496249253431930, −6.52413767790166507239918565829, −5.68658093344747612000384361061, −5.43834558561914446701793062931, −3.55512960583277148514522069155, −3.04368999033989255146561015606, −0.72005684250205394920292711542,
1.11615921333200950098986118551, 2.36433346721932089537466772419, 3.22096169424883781089225150838, 4.63228125214806638642787332323, 5.78884180274510819057945953671, 6.72132967873692842662758617850, 7.29278432444615078524794767476, 9.011115001727407044883185273348, 9.419367885945565371775249197431, 9.981874640036896128445482430683