L(s) = 1 | + 1.73i·3-s + (1.5 − 2.59i)5-s − 2.99·9-s + (−1.5 − 2.59i)11-s + (−0.5 + 0.866i)13-s + (4.5 + 2.59i)15-s − 6·17-s + 4·19-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s − 5.19i·27-s + (−1.5 − 2.59i)29-s + (2.5 − 4.33i)31-s + (4.5 − 2.59i)33-s + 2·37-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (0.670 − 1.16i)5-s − 0.999·9-s + (−0.452 − 0.783i)11-s + (−0.138 + 0.240i)13-s + (1.16 + 0.670i)15-s − 1.45·17-s + 0.917·19-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s − 0.999i·27-s + (−0.278 − 0.482i)29-s + (0.449 − 0.777i)31-s + (0.783 − 0.452i)33-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.264503307\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.264503307\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.180235067807233258392473198288, −8.613516054753951119153771895427, −7.82966258576222510353527228687, −6.44467371891122426906962703035, −5.70652880416521864723764296952, −4.93140482931564978210486264923, −4.37857376437642729786415796441, −3.19645661435219207931314840825, −2.08540843382178549723940379934, −0.45163471260014307098992752618,
1.51480534531772980035154622036, 2.52625691503245215099659752697, 3.11194513590526314294780090565, 4.67524529839879655109708350273, 5.66442290022989852770416551410, 6.44533163813515332751528408770, 7.10023790597701166111981393015, 7.58810735598572486100210286387, 8.633322227471780737801032985010, 9.511250850547524667658744975281