| L(s) = 1 | + (−1.32 + 0.499i)2-s + (−0.940 − 1.13i)3-s + (1.50 − 1.32i)4-s + (1.84 + 1.25i)5-s + (1.81 + 1.03i)6-s + (2.76 − 0.898i)7-s + (−1.32 + 2.49i)8-s + (0.154 − 0.809i)9-s + (−3.07 − 0.740i)10-s + (0.200 − 1.58i)11-s + (−2.91 − 0.463i)12-s + (1.06 − 5.58i)13-s + (−3.21 + 2.57i)14-s + (−0.308 − 3.28i)15-s + (0.506 − 3.96i)16-s + (−0.202 − 0.790i)17-s + ⋯ |
| L(s) = 1 | + (−0.935 + 0.353i)2-s + (−0.542 − 0.656i)3-s + (0.750 − 0.660i)4-s + (0.826 + 0.562i)5-s + (0.739 + 0.422i)6-s + (1.04 − 0.339i)7-s + (−0.468 + 0.883i)8-s + (0.0514 − 0.269i)9-s + (−0.972 − 0.234i)10-s + (0.0605 − 0.479i)11-s + (−0.841 − 0.133i)12-s + (0.295 − 1.54i)13-s + (−0.858 + 0.687i)14-s + (−0.0796 − 0.847i)15-s + (0.126 − 0.991i)16-s + (−0.0492 − 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0822 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0822 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.744305 - 0.685420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.744305 - 0.685420i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 - 0.499i)T \) |
| 5 | \( 1 + (-1.84 - 1.25i)T \) |
| good | 3 | \( 1 + (0.940 + 1.13i)T + (-0.562 + 2.94i)T^{2} \) |
| 7 | \( 1 + (-2.76 + 0.898i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.200 + 1.58i)T + (-10.6 - 2.73i)T^{2} \) |
| 13 | \( 1 + (-1.06 + 5.58i)T + (-12.0 - 4.78i)T^{2} \) |
| 17 | \( 1 + (0.202 + 0.790i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (5.56 + 4.60i)T + (3.56 + 18.6i)T^{2} \) |
| 23 | \( 1 + (4.00 - 4.26i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-5.00 + 0.314i)T + (28.7 - 3.63i)T^{2} \) |
| 31 | \( 1 + (-7.01 + 1.80i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (-0.512 - 0.281i)T + (19.8 + 31.2i)T^{2} \) |
| 41 | \( 1 + (-2.74 + 2.57i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (5.85 - 4.25i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (3.43 + 8.67i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (7.00 - 11.0i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (1.59 - 0.751i)T + (37.6 - 45.4i)T^{2} \) |
| 61 | \( 1 + (-6.63 + 7.06i)T + (-3.83 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.226 - 3.59i)T + (-66.4 - 8.39i)T^{2} \) |
| 71 | \( 1 + (-0.727 + 0.287i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (8.00 + 3.76i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (7.74 + 9.36i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (-3.43 + 4.15i)T + (-15.5 - 81.5i)T^{2} \) |
| 89 | \( 1 + (-1.41 + 3.00i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-9.58 + 0.603i)T + (96.2 - 12.1i)T^{2} \) |
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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
−9.918549127169125492138978762272, −8.805086899579641170451291540309, −8.044337194684294296111346203227, −7.30135607032022271677860871292, −6.33693841227438163586922728307, −5.94554228800248753648858905614, −4.88631049181459730134366555090, −3.01367983425285473634507951147, −1.78368557739903234859394872798, −0.68748208853336276293746741629,
1.62105003718852856603682538005, 2.20670298091214837438609906218, 4.24469398565874704729785889344, 4.71814933694292830338671882617, 6.03433642207291920124975541127, 6.67885890209079828200062732229, 8.294134764956899437232928310342, 8.371027820022397489925451862602, 9.542657998290411048729203961850, 10.13290317843787834109616609022
