L(s) = 1 | + i·2-s + 1.56i·3-s − 4-s − 1.56·6-s − 1.56i·7-s − i·8-s + 0.561·9-s + 4·11-s − 1.56i·12-s − 6.68i·13-s + 1.56·14-s + 16-s − 7.56i·17-s + 0.561i·18-s + 19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.901i·3-s − 0.5·4-s − 0.637·6-s − 0.590i·7-s − 0.353i·8-s + 0.187·9-s + 1.20·11-s − 0.450i·12-s − 1.85i·13-s + 0.417·14-s + 0.250·16-s − 1.83i·17-s + 0.132i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52100 + 0.359060i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52100 + 0.359060i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.56iT - 3T^{2} \) |
| 7 | \( 1 + 1.56iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6.68iT - 13T^{2} \) |
| 17 | \( 1 + 7.56iT - 17T^{2} \) |
| 23 | \( 1 + 4.68iT - 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 + 0.438iT - 53T^{2} \) |
| 59 | \( 1 - 1.56T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 1.56iT - 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 - 10.6iT - 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−9.969119765412214412609468782456, −9.355595615722813378262012642674, −8.473050134023461523107696118017, −7.43582938200106698202068195487, −6.84959769768094511393835319048, −5.63707008418649172748193183955, −4.85734995906847119650561445981, −4.00275482205706600790667209027, −3.06319225126066391878135551083, −0.817839717350066967349898810547,
1.54407781671307853647558981084, 1.95037387115377977820933789498, 3.66794297247418809718432623874, 4.35049974592388618370427717040, 5.83488378895036279430932122721, 6.55742262621085631138954296499, 7.40542089656531297117585970930, 8.500346192139348189090935184619, 9.177541710257951694811382740348, 9.866700187967976097811467289444