L(s) = 1 | + (−1.34 − 0.450i)2-s + (1.59 + 1.20i)4-s + (−0.254 − 2.22i)5-s − 2.64i·7-s + (−1.59 − 2.33i)8-s + (−0.659 + 3.09i)10-s + 1.51i·11-s − 3.87·13-s + (−1.18 + 3.54i)14-s + (1.08 + 3.84i)16-s + 3.31i·17-s − 7.08i·19-s + (2.27 − 3.84i)20-s + (0.681 − 2.02i)22-s − 4.82i·23-s + ⋯ |
L(s) = 1 | + (−0.947 − 0.318i)2-s + (0.797 + 0.603i)4-s + (−0.113 − 0.993i)5-s − 0.998i·7-s + (−0.563 − 0.825i)8-s + (−0.208 + 0.978i)10-s + 0.456i·11-s − 1.07·13-s + (−0.317 + 0.946i)14-s + (0.271 + 0.962i)16-s + 0.803i·17-s − 1.62i·19-s + (0.508 − 0.860i)20-s + (0.145 − 0.432i)22-s − 1.00i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.213960 - 0.574608i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.213960 - 0.574608i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.34 + 0.450i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.254 + 2.22i)T \) |
good | 7 | \( 1 + 2.64iT - 7T^{2} \) |
| 11 | \( 1 - 1.51iT - 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 - 3.31iT - 17T^{2} \) |
| 19 | \( 1 + 7.08iT - 19T^{2} \) |
| 23 | \( 1 + 4.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.18iT - 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 - 7.87T + 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 47 | \( 1 + 7.08iT - 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 + 6.79iT - 59T^{2} \) |
| 61 | \( 1 + 3.60iT - 61T^{2} \) |
| 67 | \( 1 - 1.01T + 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 - 15.5iT - 73T^{2} \) |
| 79 | \( 1 - 7.36T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−10.94643553380002670134004195152, −10.06167680048668609592932288459, −9.307855804701985055095155880850, −8.381816148334714561617514035303, −7.47595526215678141315103767319, −6.67200517114735308651846480332, −4.97263153939161633294388923312, −3.90790894059386401818841036155, −2.17787959758663038483908588786, −0.54231708773325161022118296356,
2.09825786602457096527844522366, 3.24594377001841507187673508350, 5.35936636704649422525182040043, 6.14993483322577204668870197979, 7.28906325101535566750532260421, 7.937422511441649026993979736753, 9.141063430368802955869455135676, 9.830176914996988645803886296250, 10.74113660055816399122021222961, 11.66883007554961916058949223583