L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.0752 + 0.130i)3-s + (−0.499 − 0.866i)4-s + (−1.87 + 1.22i)5-s − 0.150·6-s + (−2.53 − 0.742i)7-s + 0.999·8-s + (1.48 − 2.57i)9-s + (−0.125 − 2.23i)10-s + (2.31 + 2.37i)11-s + (0.0752 − 0.130i)12-s − 2.51i·13-s + (1.91 − 1.82i)14-s + (−0.300 − 0.151i)15-s + (−0.5 + 0.866i)16-s + (−1.99 + 1.14i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.0434 + 0.0752i)3-s + (−0.249 − 0.433i)4-s + (−0.836 + 0.547i)5-s − 0.0614·6-s + (−0.959 − 0.280i)7-s + 0.353·8-s + (0.496 − 0.859i)9-s + (−0.0396 − 0.705i)10-s + (0.697 + 0.716i)11-s + (0.0217 − 0.0376i)12-s − 0.697i·13-s + (0.511 − 0.488i)14-s + (−0.0775 − 0.0391i)15-s + (−0.125 + 0.216i)16-s + (−0.482 + 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.975611 + 0.206624i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.975611 + 0.206624i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (1.87 - 1.22i)T \) |
| 7 | \( 1 + (2.53 + 0.742i)T \) |
| 11 | \( 1 + (-2.31 - 2.37i)T \) |
good | 3 | \( 1 + (-0.0752 - 0.130i)T + (-1.5 + 2.59i)T^{2} \) |
| 13 | \( 1 + 2.51iT - 13T^{2} \) |
| 17 | \( 1 + (1.99 - 1.14i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.16 + 5.47i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.94 - 2.85i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.85iT - 29T^{2} \) |
| 31 | \( 1 + (-1.82 + 1.05i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.79 - 5.65i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.710T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 + (-3.34 + 5.78i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.78 + 2.18i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.75 + 4.47i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.17 + 8.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.84 - 3.94i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + (-11.5 + 6.65i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.29 - 4.21i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8.40iT - 83T^{2} \) |
| 89 | \( 1 + (-3.02 - 1.74i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
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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
−10.14747662862609966740405795401, −9.478483506504017934805542461360, −8.759265990019860154511915804275, −7.52772261191996650536185295128, −6.88861386329408130328123547825, −6.46183512369715428739155503360, −4.96653153800866398370425172541, −3.89544776023166980313605190168, −3.02511824688570848551667375788, −0.800571076746694962802611950619,
0.986469619955417285163988179853, 2.55720427780832551470730177212, 3.76490531090289032088734454466, 4.49555697630975445771124806396, 5.83248260300184385649451945707, 6.98568325682096065055512449021, 7.83448480093543343161238908893, 8.727903911808467442484096378174, 9.358983923923718108058697332331, 10.21438467444218824577742424661