L(s) = 1 | + 2.43·3-s + 3.70i·5-s + i·7-s + 2.95·9-s − i·11-s + (1.47 + 3.29i)13-s + 9.03i·15-s + 0.503·17-s − 0.759i·19-s + 2.43i·21-s + 7.83·23-s − 8.70·25-s − 0.114·27-s − 8.07·29-s + 3.66i·31-s + ⋯ |
L(s) = 1 | + 1.40·3-s + 1.65i·5-s + 0.377i·7-s + 0.984·9-s − 0.301i·11-s + (0.408 + 0.912i)13-s + 2.33i·15-s + 0.122·17-s − 0.174i·19-s + 0.532i·21-s + 1.63·23-s − 1.74·25-s − 0.0219·27-s − 1.50·29-s + 0.657i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.027134119\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.027134119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-1.47 - 3.29i)T \) |
good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 - 3.70iT - 5T^{2} \) |
| 17 | \( 1 - 0.503T + 17T^{2} \) |
| 19 | \( 1 + 0.759iT - 19T^{2} \) |
| 23 | \( 1 - 7.83T + 23T^{2} \) |
| 29 | \( 1 + 8.07T + 29T^{2} \) |
| 31 | \( 1 - 3.66iT - 31T^{2} \) |
| 37 | \( 1 - 5.15iT - 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 + 2.20T + 43T^{2} \) |
| 47 | \( 1 + 3.86iT - 47T^{2} \) |
| 53 | \( 1 - 7.41T + 53T^{2} \) |
| 59 | \( 1 + 3.16iT - 59T^{2} \) |
| 61 | \( 1 - 4.30T + 61T^{2} \) |
| 67 | \( 1 + 14.7iT - 67T^{2} \) |
| 71 | \( 1 - 5.64iT - 71T^{2} \) |
| 73 | \( 1 + 14.4iT - 73T^{2} \) |
| 79 | \( 1 + 3.53T + 79T^{2} \) |
| 83 | \( 1 - 8.51iT - 83T^{2} \) |
| 89 | \( 1 + 13.3iT - 89T^{2} \) |
| 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
−8.684411583038151611518208848503, −7.985806106402633874619664094418, −7.18284417856303150768453982097, −6.74537967523022621969848355629, −5.93832801924192494129450690051, −4.78743047966432161416424834712, −3.59764196712166175110742881056, −3.23839821561361199027752574243, −2.51873383641577025127455993445, −1.67307536543779307174350580812,
0.69349062186293089139878562972, 1.64819333201782656764937585932, 2.64039990867189308977323271940, 3.71102430265703759334691635400, 4.15730428305469291123610317342, 5.25322243362815150468226720687, 5.68947032349815069384855465658, 7.19110994761504050644048206586, 7.60795194435808551104648420485, 8.443507195216155402615009929364