L(s) = 1 | + (1.06 + 0.933i)2-s + (−0.707 + 0.707i)3-s + (0.256 + 1.98i)4-s + (−1.24 + 1.86i)5-s + (−1.41 + 0.0909i)6-s − 1.58·7-s + (−1.57 + 2.34i)8-s − 1.00i·9-s + (−3.05 + 0.817i)10-s + (3.92 − 3.92i)11-s + (−1.58 − 1.22i)12-s + (−3.10 + 3.10i)13-s + (−1.68 − 1.48i)14-s + (−0.438 − 2.19i)15-s + (−3.86 + 1.01i)16-s + 1.48i·17-s + ⋯ |
L(s) = 1 | + (0.751 + 0.660i)2-s + (−0.408 + 0.408i)3-s + (0.128 + 0.991i)4-s + (−0.554 + 0.831i)5-s + (−0.576 + 0.0371i)6-s − 0.600·7-s + (−0.558 + 0.829i)8-s − 0.333i·9-s + (−0.966 + 0.258i)10-s + (1.18 − 1.18i)11-s + (−0.457 − 0.352i)12-s + (−0.861 + 0.861i)13-s + (−0.451 − 0.396i)14-s + (−0.113 − 0.566i)15-s + (−0.967 + 0.254i)16-s + 0.359i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.477665 + 1.26226i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.477665 + 1.26226i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.06 - 0.933i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.24 - 1.86i)T \) |
good | 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 + (-3.92 + 3.92i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.10 - 3.10i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.48iT - 17T^{2} \) |
| 19 | \( 1 + (-4.94 - 4.94i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 + (-4.42 - 4.42i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 + (-2.14 - 2.14i)T + 37iT^{2} \) |
| 41 | \( 1 + 6.84iT - 41T^{2} \) |
| 43 | \( 1 + (0.322 + 0.322i)T + 43iT^{2} \) |
| 47 | \( 1 + 13.3iT - 47T^{2} \) |
| 53 | \( 1 + (-0.931 - 0.931i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.14 + 1.14i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.67 - 2.67i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.43 + 5.43i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.26iT - 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 + 6.52T + 79T^{2} \) |
| 83 | \( 1 + (-0.973 + 0.973i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.83iT - 89T^{2} \) |
| 97 | \( 1 - 15.8iT - 97T^{2} \) |
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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
−12.27377031114936411894655812979, −11.75599846009369233568074188588, −10.87186050520810977601728037129, −9.553130064873673466633968512276, −8.466891819725378043966709880437, −7.06811628069127550787811502977, −6.53048597808373829805687979459, −5.37239779501274283830268981800, −3.92793805290246552421129217967, −3.19871450923356067318019036834,
0.981442168992330466032825982912, 2.90628014638010577373056192735, 4.46638819864905287185369439767, 5.19192704526570166093668906424, 6.61903692624650942139947606468, 7.49720751936344345720208730735, 9.264163897260879477789640798449, 9.791447931472281080944002739557, 11.26623577117612202976814780740, 11.89953804454792918058243811501