L(s) = 1 | + (1.5 − 2.59i)3-s + (−2 + 3.46i)4-s + (−6.5 + 2.59i)7-s + (−4.5 − 7.79i)9-s + (6 + 10.3i)12-s + 23·13-s + (−7.99 − 13.8i)16-s + (−5.5 − 9.52i)19-s + (−3 + 20.7i)21-s + (−12.5 + 21.6i)25-s − 27·27-s + (4 − 27.7i)28-s + (6.5 − 11.2i)31-s + 36·36-s + (36.5 + 63.2i)37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.928 + 0.371i)7-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)12-s + 1.76·13-s + (−0.499 − 0.866i)16-s + (−0.289 − 0.501i)19-s + (−0.142 + 0.989i)21-s + (−0.5 + 0.866i)25-s − 27-s + (0.142 − 0.989i)28-s + (0.209 − 0.363i)31-s + 36-s + (0.986 + 1.70i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.877856 - 0.0910609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.877856 - 0.0910609i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 + 2.59i)T \) |
| 7 | \( 1 + (6.5 - 2.59i)T \) |
good | 2 | \( 1 + (2 - 3.46i)T^{2} \) |
| 5 | \( 1 + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 23T + 169T^{2} \) |
| 17 | \( 1 + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (5.5 + 9.52i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + (-6.5 + 11.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-36.5 - 63.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 61T + 1.84e3T^{2} \) |
| 47 | \( 1 + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (37 + 64.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-6.5 + 11.2i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + (-48.5 + 84.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−18.18469278944139048887368297192, −16.84208441965893813213973103800, −15.42721410848876837977762970691, −13.54581719826265743074080423457, −13.05608419011204828296859916744, −11.67230751947332469346862091920, −9.252588891088115612641962602212, −8.157996158947256526575565289898, −6.45358831043530219069890543980, −3.36578704399175758378478904088,
3.95940027798559613597211892625, 6.00577066649538924854715699344, 8.616907772579954865039179977931, 9.853380481351000633855114089440, 10.85867357147380919384377244288, 13.25824231990411862652613133590, 14.19262124482623320311386052803, 15.53367582927521313680246387256, 16.41480989137551148618680286450, 18.22591168197302180246445512932