L(s) = 1 | + (−1.34 − 0.430i)2-s + (0.916 − 1.46i)3-s + (1.62 + 1.15i)4-s − 0.348i·5-s + (−1.86 + 1.58i)6-s − i·7-s + (−1.69 − 2.26i)8-s + (−1.31 − 2.69i)9-s + (−0.150 + 0.469i)10-s + 3.90·11-s + (3.19 − 1.33i)12-s − 2.93·13-s + (−0.430 + 1.34i)14-s + (−0.512 − 0.319i)15-s + (1.30 + 3.77i)16-s + 3.90i·17-s + ⋯ |
L(s) = 1 | + (−0.952 − 0.304i)2-s + (0.529 − 0.848i)3-s + (0.814 + 0.579i)4-s − 0.155i·5-s + (−0.762 + 0.647i)6-s − 0.377i·7-s + (−0.599 − 0.800i)8-s + (−0.439 − 0.898i)9-s + (−0.0474 + 0.148i)10-s + 1.17·11-s + (0.923 − 0.384i)12-s − 0.815·13-s + (−0.115 + 0.360i)14-s + (−0.132 − 0.0825i)15-s + (0.327 + 0.944i)16-s + 0.946i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.634652 - 0.423224i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.634652 - 0.423224i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.34 + 0.430i)T \) |
| 3 | \( 1 + (-0.916 + 1.46i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 0.348iT - 5T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 - 3.90iT - 17T^{2} \) |
| 19 | \( 1 - 5.57iT - 19T^{2} \) |
| 23 | \( 1 + 2.18T + 23T^{2} \) |
| 29 | \( 1 - 9.75iT - 29T^{2} \) |
| 31 | \( 1 + 2.63iT - 31T^{2} \) |
| 37 | \( 1 - 0.639T + 37T^{2} \) |
| 41 | \( 1 + 7.57iT - 41T^{2} \) |
| 43 | \( 1 + 2.51iT - 43T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 53 | \( 1 + 1.72iT - 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 0.639iT - 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 7.87T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + 10.5iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−14.15443601066916394832959918017, −12.59275088346943774373586210846, −12.13501588647665041178072148572, −10.70136724791868755874645547442, −9.444454736041171823751338318279, −8.476520618550882199583163800579, −7.39747215081505625789662935432, −6.35811556016261333179859808146, −3.57684775181028715928786402794, −1.63003952872010285938024072312,
2.69342734417934637179760498892, 4.84079099299623635736196969914, 6.54630502783335092360923247119, 7.900853571548419256767241269030, 9.186085137699451434554357060176, 9.644179884422761290962469808312, 11.00360802506305194577379649507, 11.91211796666657047285241328469, 13.86522840398584556303789378407, 14.80557677603276154933193375641