L(s) = 1 | + (−1.39 − 1.02i)3-s + (−0.667 + 1.15i)5-s + (−1.90 − 1.83i)7-s + (0.880 + 2.86i)9-s + (0.756 + 1.31i)11-s + (−2.58 − 4.48i)13-s + (2.11 − 0.923i)15-s + (0.774 − 1.34i)17-s + (1.25 + 2.16i)19-s + (0.757 + 4.51i)21-s + (−3.68 + 6.37i)23-s + (1.60 + 2.78i)25-s + (1.72 − 4.90i)27-s + (−0.0309 + 0.0536i)29-s + 3.84·31-s + ⋯ |
L(s) = 1 | + (−0.804 − 0.594i)3-s + (−0.298 + 0.516i)5-s + (−0.719 − 0.694i)7-s + (0.293 + 0.955i)9-s + (0.228 + 0.395i)11-s + (−0.717 − 1.24i)13-s + (0.547 − 0.238i)15-s + (0.187 − 0.325i)17-s + (0.287 + 0.497i)19-s + (0.165 + 0.986i)21-s + (−0.767 + 1.32i)23-s + (0.321 + 0.557i)25-s + (0.332 − 0.943i)27-s + (−0.00575 + 0.00996i)29-s + 0.691·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7415745423\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7415745423\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.39 + 1.02i)T \) |
| 7 | \( 1 + (1.90 + 1.83i)T \) |
good | 5 | \( 1 + (0.667 - 1.15i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.756 - 1.31i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.58 + 4.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.774 + 1.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 2.16i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.68 - 6.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0309 - 0.0536i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.51 - 7.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.09 - 8.83i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 9.51T + 47T^{2} \) |
| 53 | \( 1 + (-0.755 + 1.30i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 8.44T + 59T^{2} \) |
| 61 | \( 1 - 3.23T + 61T^{2} \) |
| 67 | \( 1 + 6.93T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 5.91T + 79T^{2} \) |
| 83 | \( 1 + (2.80 - 4.85i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.703 - 1.21i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.09 - 10.5i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.06854484904462968083941047326, −9.649161092784018867712028482452, −7.955930377964472220580014867216, −7.51421402853889710223759949767, −6.77744076736908785540620990310, −5.89383725930807359518902255640, −5.01268855573577131890349187019, −3.78019158931629649391936300260, −2.72010998386897578700393255373, −1.06580315227596789057935993124,
0.45606050691765940457999696647, 2.39756355375532837165302699089, 3.82165879370733583904353931482, 4.56824732092182528335606035773, 5.51238002511966042802035405553, 6.37649666554423730990808501038, 7.05081413703242354236112658475, 8.539077366409366680976931607028, 9.036068661404697476325697611582, 9.876511343646890748359664693281