| L(s) = 1 | + (0.921 + 1.07i)2-s + (−0.964 − 1.43i)3-s + (−0.300 + 1.97i)4-s + (−2.14 − 0.638i)5-s + (0.653 − 2.36i)6-s − 4.06·7-s + (−2.39 + 1.50i)8-s + (−1.13 + 2.77i)9-s + (−1.29 − 2.88i)10-s + (2.61 + 1.89i)11-s + (3.13 − 1.47i)12-s + (−3.55 − 4.88i)13-s + (−3.74 − 4.36i)14-s + (1.14 + 3.69i)15-s + (−3.81 − 1.18i)16-s + (−0.528 + 1.62i)17-s + ⋯ |
| L(s) = 1 | + (0.651 + 0.758i)2-s + (−0.556 − 0.830i)3-s + (−0.150 + 0.988i)4-s + (−0.958 − 0.285i)5-s + (0.266 − 0.963i)6-s − 1.53·7-s + (−0.847 + 0.530i)8-s + (−0.379 + 0.925i)9-s + (−0.408 − 0.912i)10-s + (0.787 + 0.571i)11-s + (0.904 − 0.425i)12-s + (−0.985 − 1.35i)13-s + (−1.00 − 1.16i)14-s + (0.296 + 0.954i)15-s + (−0.954 − 0.297i)16-s + (−0.128 + 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0133795 - 0.0727929i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0133795 - 0.0727929i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.921 - 1.07i)T \) |
| 3 | \( 1 + (0.964 + 1.43i)T \) |
| 5 | \( 1 + (2.14 + 0.638i)T \) |
| good | 7 | \( 1 + 4.06T + 7T^{2} \) |
| 11 | \( 1 + (-2.61 - 1.89i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (3.55 + 4.88i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.528 - 1.62i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.49 + 0.809i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.71 - 3.74i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.01 + 0.655i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.77 - 0.576i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.36 + 1.88i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.31 + 5.94i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 5.20T + 43T^{2} \) |
| 47 | \( 1 + (3.27 - 1.06i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.20 - 6.78i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (9.81 - 7.13i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.99 + 3.62i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.983 - 3.02i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.67 + 5.15i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.58 - 2.17i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.42 - 0.787i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.77 - 1.87i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (4.38 - 6.04i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (14.9 - 4.85i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.38650800057571725271550136293, −11.96658717904202880753126113302, −10.54390975338197083106517581570, −9.211252191062801064113976061062, −8.007512666633556627866210463686, −7.24587858230394592487704822241, −6.48049404061631087796460848495, −5.48276494122232443624588209551, −4.19694682125746904969244842549, −2.93666897711148886661962172550,
0.04457734544900589500261942674, 2.94873517009210107555430361223, 3.91121089348413458955834251792, 4.67017138045140727364110209378, 6.30796343670869060369595983839, 6.71513087397995381743356816899, 8.851220744131786611139570646866, 9.644811088327113283579857641768, 10.38522230557982562451792058408, 11.44103101356574929751055291537
