L(s) = 1 | + (−2.22 − 1.28i)3-s − 3.56i·7-s + (1.79 + 3.11i)9-s + 4.56·11-s + (−0.866 + 0.5i)13-s + (4.96 + 2.86i)17-s + (4.35 − 0.221i)19-s + (−4.58 + 7.93i)21-s + (−4.84 + 2.79i)23-s − 1.53i·27-s + (3.38 + 5.86i)29-s + 4.59·31-s + (−10.1 − 5.86i)33-s − 3.56i·37-s + 2.56·39-s + ⋯ |
L(s) = 1 | + (−1.28 − 0.741i)3-s − 1.34i·7-s + (0.599 + 1.03i)9-s + 1.37·11-s + (−0.240 + 0.138i)13-s + (1.20 + 0.695i)17-s + (0.998 − 0.0507i)19-s + (−1.00 + 1.73i)21-s + (−1.01 + 0.583i)23-s − 0.296i·27-s + (0.628 + 1.08i)29-s + 0.826·31-s + (−1.76 − 1.02i)33-s − 0.586i·37-s + 0.411·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.246652438\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.246652438\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4.35 + 0.221i)T \) |
good | 3 | \( 1 + (2.22 + 1.28i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 3.56iT - 7T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.96 - 2.86i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (4.84 - 2.79i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.38 - 5.86i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 37 | \( 1 + 3.56iT - 37T^{2} \) |
| 41 | \( 1 + (5.35 - 9.27i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.79 - 5.65i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.86 + 3.38i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.94 - 1.70i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.38 - 5.86i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.06 + 7.04i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.48 + 3.16i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.515 + 0.892i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 6.36i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.43 + 4.21i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.40iT - 83T^{2} \) |
| 89 | \( 1 + (-6.13 - 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.27 - 1.31i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.289645698113426596749337664562, −7.945233667275518591853255175899, −7.42758591740184701589399621547, −6.63366245574250942053969633868, −6.12494819343800829009205537793, −5.18867803539148166800756219623, −4.21826914311819954352114443159, −3.38249511460576798057836387804, −1.44471952253781392174823723220, −0.931060673336458747354398034377,
0.825335686299070068595831762352, 2.42336055382173869414259779449, 3.64637853776994842169067742646, 4.59058207489195147280807345452, 5.42704168059648798718227126793, 5.91702980611598161726390765710, 6.62506952552264526421190906667, 7.75556884743596127934376681812, 8.732953063721960163153645291097, 9.567178959229187963330123120978