L(s) = 1 | + (−0.707 − 0.707i)3-s + (−1.95 − 1.07i)5-s + 1.22·7-s + 1.00i·9-s + (−1.38 − 1.38i)11-s + (2.12 + 2.12i)13-s + (0.623 + 2.14i)15-s − 6.00i·17-s + (3.06 − 3.06i)19-s + (−0.863 − 0.863i)21-s − 2.90·23-s + (2.67 + 4.22i)25-s + (0.707 − 0.707i)27-s + (−3.18 + 3.18i)29-s − 3.88·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.876 − 0.481i)5-s + 0.461·7-s + 0.333i·9-s + (−0.416 − 0.416i)11-s + (0.588 + 0.588i)13-s + (0.161 + 0.554i)15-s − 1.45i·17-s + (0.702 − 0.702i)19-s + (−0.188 − 0.188i)21-s − 0.606·23-s + (0.535 + 0.844i)25-s + (0.136 − 0.136i)27-s + (−0.591 + 0.591i)29-s − 0.697·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5683294935\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5683294935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.95 + 1.07i)T \) |
good | 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 + (1.38 + 1.38i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.12 - 2.12i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.00iT - 17T^{2} \) |
| 19 | \( 1 + (-3.06 + 3.06i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 + (3.18 - 3.18i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.38iT - 41T^{2} \) |
| 43 | \( 1 + (-9.00 + 9.00i)T - 43iT^{2} \) |
| 47 | \( 1 + 0.586iT - 47T^{2} \) |
| 53 | \( 1 + (2.36 - 2.36i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.43 + 8.43i)T + 59iT^{2} \) |
| 61 | \( 1 + (9.98 - 9.98i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.82 + 3.82i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 + 1.31T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + (2.91 + 2.91i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.58iT - 89T^{2} \) |
| 97 | \( 1 - 9.45iT - 97T^{2} \) |
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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
−8.963341891444420909683947768965, −7.69599024360163676974378248842, −7.56809280567894325529038312957, −6.51758833247607519013035374761, −5.45309015125255996682272239308, −4.86367368382941125534395960260, −3.91516241254212984992005963242, −2.82044529437564836681970880011, −1.42394031938062166603530071690, −0.23289231026891852375627822402,
1.50849720038716162012695511576, 2.99298170348410485762295143708, 3.90119287731382299112777642117, 4.52483391408590451953685440250, 5.70684502476484810839286884336, 6.20776867552056126970659516461, 7.48592593957182896794412863831, 7.88720390076671517369714605908, 8.650073773587353404031930933783, 9.785484008984343056900911218081