L(s) = 1 | − 1.10·2-s + (−1.22 + 1.22i)3-s − 0.783·4-s + (−0.0527 + 0.0913i)5-s + (1.34 − 1.35i)6-s + 3.07·8-s + (−0.0231 − 2.99i)9-s + (0.0581 − 0.100i)10-s + (−1.66 − 2.89i)11-s + (0.956 − 0.963i)12-s + (−1.23 − 2.14i)13-s + (−0.0479 − 0.176i)15-s − 1.81·16-s + (−0.806 + 1.39i)17-s + (0.0255 + 3.30i)18-s + (3.84 + 6.65i)19-s + ⋯ |
L(s) = 1 | − 0.779·2-s + (−0.704 + 0.709i)3-s − 0.391·4-s + (−0.0235 + 0.0408i)5-s + (0.549 − 0.553i)6-s + 1.08·8-s + (−0.00772 − 0.999i)9-s + (0.0183 − 0.0318i)10-s + (−0.503 − 0.871i)11-s + (0.276 − 0.278i)12-s + (−0.343 − 0.595i)13-s + (−0.0123 − 0.0455i)15-s − 0.454·16-s + (−0.195 + 0.338i)17-s + (0.00602 + 0.779i)18-s + (0.881 + 1.52i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.570843 + 0.124483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.570843 + 0.124483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.22 - 1.22i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.10T + 2T^{2} \) |
| 5 | \( 1 + (0.0527 - 0.0913i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.66 + 2.89i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.23 + 2.14i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.806 - 1.39i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.84 - 6.65i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.948 + 1.64i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.64 + 8.04i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.26T + 31T^{2} \) |
| 37 | \( 1 + (-0.991 - 1.71i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.74 - 6.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.77 - 6.53i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.19T + 47T^{2} \) |
| 53 | \( 1 + (-4.98 + 8.64i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 4.45T + 59T^{2} \) |
| 61 | \( 1 + 5.67T + 61T^{2} \) |
| 67 | \( 1 - 9.97T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 + (-2.36 + 4.09i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 + (0.584 - 1.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.01 + 5.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.90 - 3.29i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.95015392843709681250339669587, −10.01267515516036057802196235067, −9.765089887931507504054176409743, −8.420585959026046866200970292632, −7.912815615899584683511210927164, −6.40470953747152257728877014268, −5.41316537396468239912058948354, −4.48605562568878583029367141046, −3.22759772254401739002411378053, −0.848367774066355131763228215115,
0.855388311194051926557906523150, 2.43329831352209337251276493718, 4.62377071645734803476064546134, 5.14508747804976273133422060479, 6.82082746241027914354828788585, 7.27196519114016874069842740293, 8.343807401125866400035774957580, 9.243461209518037518191865863374, 10.17060573473639250144316638505, 10.91891472365187781948794973891