L(s) = 1 | − 1.41i·2-s + (0.923 − 0.382i)3-s − 1.00·4-s + (−0.541 − 1.30i)6-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)12-s − 0.765i·13-s − 0.999·16-s + (−1.00 − i)18-s − i·23-s + 25-s − 1.08·26-s + (0.382 − 0.923i)27-s + 1.84i·31-s + 1.41i·32-s + ⋯ |
L(s) = 1 | − 1.41i·2-s + (0.923 − 0.382i)3-s − 1.00·4-s + (−0.541 − 1.30i)6-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)12-s − 0.765i·13-s − 0.999·16-s + (−1.00 − i)18-s − i·23-s + 25-s − 1.08·26-s + (0.382 − 0.923i)27-s + 1.84i·31-s + 1.41i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.734601990\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.734601990\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.923 + 0.382i)T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 0.765iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.84iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.765T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.765T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.84T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 - 1.84iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.753228811183593212910275181989, −7.921061635016528623060460266361, −7.01975949435002868172316675109, −6.40246504438196727514117995697, −5.05934392444512838441553851025, −4.26160364775161721727916737979, −3.25213482432495438759603118624, −2.89610536498572168842851833290, −1.90244534019437598780644814806, −0.960705407554033885350667234903,
1.80508170378771722012171428637, 2.83301132027076209522150946685, 3.94458697219148524002949471647, 4.63006185745894415018139296042, 5.42498502380004612584396979831, 6.28240433635837408768435482282, 7.05963629361279602150327858637, 7.64526836111213968397864204174, 8.216882095027301463680511534515, 9.086701100533405658269068786721