L(s) = 1 | + (−0.991 + 0.130i)2-s + (0.965 − 0.258i)4-s + (−0.389 + 1.95i)5-s + (−0.923 + 0.382i)8-s + (0.793 − 0.608i)9-s + (0.130 − 1.99i)10-s + (−0.608 + 0.793i)13-s + (0.866 − 0.5i)16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)18-s + (0.130 + 1.99i)20-s + (−2.75 − 1.14i)25-s + (0.499 − 0.866i)26-s + (−0.391 + 0.793i)29-s + (−0.793 + 0.608i)32-s + ⋯ |
L(s) = 1 | + (−0.991 + 0.130i)2-s + (0.965 − 0.258i)4-s + (−0.389 + 1.95i)5-s + (−0.923 + 0.382i)8-s + (0.793 − 0.608i)9-s + (0.130 − 1.99i)10-s + (−0.608 + 0.793i)13-s + (0.866 − 0.5i)16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)18-s + (0.130 + 1.99i)20-s + (−2.75 − 1.14i)25-s + (0.499 − 0.866i)26-s + (−0.391 + 0.793i)29-s + (−0.793 + 0.608i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5857823166\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5857823166\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.991 - 0.130i)T \) |
| 13 | \( 1 + (0.608 - 0.793i)T \) |
| 17 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.793 + 0.608i)T^{2} \) |
| 5 | \( 1 + (0.389 - 1.95i)T + (-0.923 - 0.382i)T^{2} \) |
| 7 | \( 1 + (-0.793 - 0.608i)T^{2} \) |
| 11 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 19 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 23 | \( 1 + (0.130 - 0.991i)T^{2} \) |
| 29 | \( 1 + (0.391 - 0.793i)T + (-0.608 - 0.793i)T^{2} \) |
| 31 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 37 | \( 1 + (1.34 + 0.665i)T + (0.608 + 0.793i)T^{2} \) |
| 41 | \( 1 + (-0.423 - 0.483i)T + (-0.130 + 0.991i)T^{2} \) |
| 43 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.241 - 0.0999i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 61 | \( 1 + (-0.583 - 1.18i)T + (-0.608 + 0.793i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.991 + 0.130i)T^{2} \) |
| 73 | \( 1 + (-1.85 - 0.369i)T + (0.923 + 0.382i)T^{2} \) |
| 79 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.732 + 0.835i)T + (-0.130 - 0.991i)T^{2} \) |
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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
−10.39333208880432701455724181762, −9.935463307612917473923728303998, −9.024937515153767750643776516216, −7.81571070173819460599840516564, −7.14354167485048618751378991780, −6.72457544236381875194786867444, −5.80032478628908633980869789083, −3.96079925221905127490250940483, −3.06038450425419013327317212404, −1.88728834406980217021795888065,
0.821521010873783412990451230843, 2.04856665946721987106615956922, 3.69889879025483692405536080482, 4.90597390432879235815447412161, 5.55081518930677564797399960030, 7.08832580168609819470907062592, 7.909079264322388646897797401691, 8.301753851981022385144775649601, 9.379941233825180728003275627410, 9.790002113711813747773500477350