L(s) = 1 | + (−0.604 + 1.62i)3-s + 0.533·5-s + (−2.26 − 1.96i)9-s + 3.92i·11-s + (0.116 − 0.0674i)13-s + (−0.322 + 0.866i)15-s + (2.16 + 3.74i)17-s + (1.93 + 1.11i)19-s − 1.96i·23-s − 4.71·25-s + (4.55 − 2.49i)27-s + (−5.16 − 2.98i)29-s + (−0.800 − 0.462i)31-s + (−6.37 − 2.37i)33-s + (−3.89 + 6.75i)37-s + ⋯ |
L(s) = 1 | + (−0.349 + 0.937i)3-s + 0.238·5-s + (−0.756 − 0.654i)9-s + 1.18i·11-s + (0.0324 − 0.0187i)13-s + (−0.0832 + 0.223i)15-s + (0.524 + 0.908i)17-s + (0.442 + 0.255i)19-s − 0.410i·23-s − 0.943·25-s + (0.877 − 0.480i)27-s + (−0.959 − 0.553i)29-s + (−0.143 − 0.0829i)31-s + (−1.10 − 0.413i)33-s + (−0.641 + 1.11i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9657927250\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9657927250\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.604 - 1.62i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 0.533T + 5T^{2} \) |
| 11 | \( 1 - 3.92iT - 11T^{2} \) |
| 13 | \( 1 + (-0.116 + 0.0674i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.16 - 3.74i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.93 - 1.11i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.96iT - 23T^{2} \) |
| 29 | \( 1 + (5.16 + 2.98i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.800 + 0.462i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.89 - 6.75i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.59 - 7.95i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.24 + 5.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.04 - 5.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.54 - 5.50i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.89 + 3.28i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9.35 - 5.39i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.75 - 9.97i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.22iT - 71T^{2} \) |
| 73 | \( 1 + (0.329 - 0.190i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.60 + 7.97i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.28 + 2.21i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.56 + 14.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.6 + 7.89i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.755699367463274538031098720201, −9.123613618352961739533270913628, −8.104597100962125638188572464627, −7.33461438322129056623750706544, −6.17981416452075084018734072938, −5.66311710528716370360987302763, −4.62072569572393699422992482289, −4.00862091371450835677868746035, −2.92273225790347619830869013006, −1.60067901859861158981395337608,
0.37539087278389075182465950333, 1.61318186668239508539772981759, 2.77329195206390302605074367106, 3.72421090524914338363733344740, 5.26868359656649596705072806108, 5.64146493688247734223701195179, 6.53635124958829156446419612232, 7.43427666396103479279712656591, 7.923378629555615789203527395249, 8.973936136770081864537706721337