L(s) = 1 | + (0.842 + 1.13i)2-s + (−0.218 − 1.71i)3-s + (−0.581 + 1.91i)4-s + (1.76 − 1.69i)6-s − 3.64·7-s + (−2.66 + 0.949i)8-s + (−2.90 + 0.750i)9-s − 5.07i·11-s + (3.41 + 0.581i)12-s − 1.70·13-s + (−3.06 − 4.14i)14-s + (−3.32 − 2.22i)16-s − 4.08·17-s + (−3.29 − 2.66i)18-s + 1.26·19-s + ⋯ |
L(s) = 1 | + (0.595 + 0.803i)2-s + (−0.126 − 0.992i)3-s + (−0.290 + 0.956i)4-s + (0.721 − 0.691i)6-s − 1.37·7-s + (−0.941 + 0.335i)8-s + (−0.968 + 0.250i)9-s − 1.52i·11-s + (0.985 + 0.168i)12-s − 0.473·13-s + (−0.820 − 1.10i)14-s + (−0.830 − 0.556i)16-s − 0.989·17-s + (−0.777 − 0.628i)18-s + 0.290·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.200161 - 0.397681i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.200161 - 0.397681i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.842 - 1.13i)T \) |
| 3 | \( 1 + (0.218 + 1.71i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.64T + 7T^{2} \) |
| 11 | \( 1 + 5.07iT - 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + 4.70iT - 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 - 4.86iT - 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 1.50iT - 41T^{2} \) |
| 43 | \( 1 - 3.43iT - 43T^{2} \) |
| 47 | \( 1 + 10.9iT - 47T^{2} \) |
| 53 | \( 1 - 8.87iT - 53T^{2} \) |
| 59 | \( 1 + 0.788iT - 59T^{2} \) |
| 61 | \( 1 - 0.627iT - 61T^{2} \) |
| 67 | \( 1 + 4.18iT - 67T^{2} \) |
| 71 | \( 1 + 6.21T + 71T^{2} \) |
| 73 | \( 1 + 4.21iT - 73T^{2} \) |
| 79 | \( 1 - 0.992iT - 79T^{2} \) |
| 83 | \( 1 - 7.72T + 83T^{2} \) |
| 89 | \( 1 + 11.5iT - 89T^{2} \) |
| 97 | \( 1 - 7.40iT - 97T^{2} \) |
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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
−10.45969434127409116722187970182, −8.994138744585862286594803065772, −8.548840669507507681565910143503, −7.40000516871540630207415527765, −6.55021515425759341694825811817, −6.14468223129552067576775879669, −5.06517450461868371761801495242, −3.50522497117049413794112800658, −2.69490745605049380088313652632, −0.19040740785265302950565128852,
2.29550417764970507851758942790, 3.39759463267970601711761892964, 4.29633881050451224172520023626, 5.16123624182269717795894300072, 6.20743938212200393903261589334, 7.15780178397729227695281803599, 8.907150587793103539418167387418, 9.728891204450249834161295211799, 9.909401908363238332219091914766, 10.91455525207444056786247756725