L(s) = 1 | + (−0.615 − 1.27i)2-s + (−1.73 − 0.0580i)3-s + (−1.24 + 1.56i)4-s + (1.66 − 0.958i)5-s + (0.991 + 2.24i)6-s + (0.708 + 2.54i)7-s + (2.75 + 0.619i)8-s + (2.99 + 0.200i)9-s + (−2.24 − 1.52i)10-s + (0.178 + 0.103i)11-s + (2.24 − 2.64i)12-s + (−0.960 − 0.554i)13-s + (2.80 − 2.47i)14-s + (−2.93 + 1.56i)15-s + (−0.909 − 3.89i)16-s + (4.54 − 2.62i)17-s + ⋯ |
L(s) = 1 | + (−0.435 − 0.900i)2-s + (−0.999 − 0.0335i)3-s + (−0.621 + 0.783i)4-s + (0.742 − 0.428i)5-s + (0.404 + 0.914i)6-s + (0.267 + 0.963i)7-s + (0.975 + 0.218i)8-s + (0.997 + 0.0669i)9-s + (−0.709 − 0.482i)10-s + (0.0538 + 0.0311i)11-s + (0.647 − 0.762i)12-s + (−0.266 − 0.153i)13-s + (0.750 − 0.660i)14-s + (−0.756 + 0.403i)15-s + (−0.227 − 0.973i)16-s + (1.10 − 0.636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.758313 - 0.386283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.758313 - 0.386283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.615 + 1.27i)T \) |
| 3 | \( 1 + (1.73 + 0.0580i)T \) |
| 7 | \( 1 + (-0.708 - 2.54i)T \) |
good | 5 | \( 1 + (-1.66 + 0.958i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.178 - 0.103i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.960 + 0.554i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.54 + 2.62i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.49 + 4.32i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.39 + 2.53i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.97 - 3.41i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 + (2.74 - 4.74i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.57 - 4.37i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.498 + 0.288i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 5.99T + 47T^{2} \) |
| 53 | \( 1 + (3.76 + 6.51i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 10.2iT - 61T^{2} \) |
| 67 | \( 1 - 8.46iT - 67T^{2} \) |
| 71 | \( 1 + 3.72iT - 71T^{2} \) |
| 73 | \( 1 + (3.45 - 1.99i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 17.4iT - 79T^{2} \) |
| 83 | \( 1 + (6.42 + 11.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.60 + 1.50i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.35 - 4.24i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.78151907418719721375927787807, −11.13246058025005278920831363589, −9.911705167669353673139931161086, −9.408227218359164979102939737440, −8.241972103344892748242531248097, −6.90851566357417687615913463266, −5.40681182994630195771252399471, −4.83590577196169059065118279929, −2.82079236029373356650510781079, −1.22128807216894292620012692825,
1.26609227410252643976727790979, 4.09508523914575344045232635206, 5.37349685744160078886901887153, 6.15448997750394965176249757906, 7.14027012259682705023132559935, 7.947357174455510348996018156178, 9.594710876549304199315907132649, 10.19146106120324403035952536800, 10.86287260891547231239331213367, 12.13753145682727214706623101830