L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.686 − 1.18i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s + 1.37·10-s + (2.18 − 3.78i)11-s + (−1 − 1.73i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 4.37·17-s + 5·19-s + (−0.686 + 1.18i)20-s + (2.18 + 3.78i)22-s + (−3.68 − 6.38i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.306 − 0.531i)5-s + (−0.188 + 0.327i)7-s + 0.353·8-s + 0.433·10-s + (0.659 − 1.14i)11-s + (−0.277 − 0.480i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + 1.06·17-s + 1.14·19-s + (−0.153 + 0.265i)20-s + (0.466 + 0.807i)22-s + (−0.768 − 1.33i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.996713 - 0.204789i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.996713 - 0.204789i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.686 + 1.18i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.18 + 3.78i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (3.68 + 6.38i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.37 + 2.37i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-5.18 - 8.98i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.55 - 7.89i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 2.74T + 53T^{2} \) |
| 59 | \( 1 + (-3.55 - 6.16i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.05 + 12.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.55 + 13.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 + (6.05 - 10.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.74 - 4.75i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 + (4.55 - 7.89i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.36455164675024265570475509744, −10.13793023661645728921978327955, −9.374926029770812745189207620591, −8.342269316485315092441089560446, −7.84261529258016309172135468118, −6.43584126441933528374313608425, −5.68280733791041058273165830676, −4.51131880856165925700181116171, −3.08583661968968612764015978118, −0.866087841543724400601761598973,
1.57879843614538794365131137200, 3.19346068723694222992876887276, 4.12592636405168422921552723852, 5.51855206258890072817998252812, 7.22057338555977952145283121708, 7.37209212560432781948332279121, 8.911771334827100226404133927301, 9.792344904414724789176825382712, 10.33639435740983412785178352322, 11.63723860492321696046302496073