L(s) = 1 | + i·2-s + (0.431 − 1.67i)3-s − 4-s − 5-s + (1.67 + 0.431i)6-s − i·8-s + (−2.62 − 1.44i)9-s − i·10-s + 5.13i·11-s + (−0.431 + 1.67i)12-s − 5.00i·13-s + (−0.431 + 1.67i)15-s + 16-s − 3.75·17-s + (1.44 − 2.62i)18-s + 2.69i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.249 − 0.968i)3-s − 0.5·4-s − 0.447·5-s + (0.684 + 0.176i)6-s − 0.353i·8-s + (−0.875 − 0.482i)9-s − 0.316i·10-s + 1.54i·11-s + (−0.124 + 0.484i)12-s − 1.38i·13-s + (−0.111 + 0.433i)15-s + 0.250·16-s − 0.911·17-s + (0.341 − 0.619i)18-s + 0.617i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7247924916\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7247924916\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.431 + 1.67i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5.13iT - 11T^{2} \) |
| 13 | \( 1 + 5.00iT - 13T^{2} \) |
| 17 | \( 1 + 3.75T + 17T^{2} \) |
| 19 | \( 1 - 2.69iT - 19T^{2} \) |
| 23 | \( 1 - 2.48iT - 23T^{2} \) |
| 29 | \( 1 - 6.18iT - 29T^{2} \) |
| 31 | \( 1 - 4.77iT - 31T^{2} \) |
| 37 | \( 1 - 0.0524T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 + 9.45T + 43T^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 - 1.11iT - 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 4.14iT - 61T^{2} \) |
| 67 | \( 1 + 0.896T + 67T^{2} \) |
| 71 | \( 1 - 13.4iT - 71T^{2} \) |
| 73 | \( 1 - 6.98iT - 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 + 1.37T + 83T^{2} \) |
| 89 | \( 1 + 5.34T + 89T^{2} \) |
| 97 | \( 1 - 0.633iT - 97T^{2} \) |
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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
−9.633039340958211805289563336417, −8.595885306306760511696057620000, −8.121708208852282417748379868084, −7.13999640406178074636146277361, −6.97573153492985533859466699619, −5.76221822770413095003943629042, −4.98482289985059110605960163231, −3.83994954126886145486242112605, −2.72269968830714293952660651531, −1.39373925773196660159062661108,
0.28480156149793925470495494164, 2.21103038615081307720873244817, 3.17475525787031952259295541263, 4.09805791674907264687737833790, 4.61445918620360335602500222916, 5.77161782540050918849315897426, 6.68624461146958251731185500505, 8.045132501304715012984509973468, 8.672619583490945951892892589156, 9.230754426363298746113075117588