| L(s) = 1 | + (−2.27 + 0.484i)2-s + (1.65 + 0.523i)3-s + (3.12 − 1.39i)4-s + (0.310 + 0.0659i)5-s + (−4.01 − 0.391i)6-s + (0.148 − 1.41i)7-s + (−2.68 + 1.94i)8-s + (2.45 + 1.72i)9-s − 0.738·10-s + (1.02 + 3.15i)11-s + (5.89 − 0.663i)12-s + (2.80 + 3.11i)13-s + (0.345 + 3.28i)14-s + (0.477 + 0.271i)15-s + (0.586 − 0.651i)16-s + (−1.92 − 5.93i)17-s + ⋯ |
| L(s) = 1 | + (−1.61 + 0.342i)2-s + (0.953 + 0.301i)3-s + (1.56 − 0.696i)4-s + (0.138 + 0.0294i)5-s + (−1.63 − 0.160i)6-s + (0.0560 − 0.533i)7-s + (−0.948 + 0.689i)8-s + (0.817 + 0.575i)9-s − 0.233·10-s + (0.309 + 0.951i)11-s + (1.70 − 0.191i)12-s + (0.778 + 0.865i)13-s + (0.0922 + 0.878i)14-s + (0.123 + 0.0699i)15-s + (0.146 − 0.162i)16-s + (−0.467 − 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.653256 + 0.221558i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.653256 + 0.221558i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.65 - 0.523i)T \) |
| 11 | \( 1 + (-1.02 - 3.15i)T \) |
| good | 2 | \( 1 + (2.27 - 0.484i)T + (1.82 - 0.813i)T^{2} \) |
| 5 | \( 1 + (-0.310 - 0.0659i)T + (4.56 + 2.03i)T^{2} \) |
| 7 | \( 1 + (-0.148 + 1.41i)T + (-6.84 - 1.45i)T^{2} \) |
| 13 | \( 1 + (-2.80 - 3.11i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (1.92 + 5.93i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (2.50 - 1.82i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.26 + 3.91i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.306 - 2.91i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (6.38 + 7.09i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.685 + 0.498i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.404 - 3.84i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (1.77 + 3.07i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.80 + 3.03i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-2.06 + 6.35i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (11.0 - 4.91i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-3.09 + 3.43i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (4.98 - 8.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.905 - 2.78i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.43 + 2.49i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.30 + 0.703i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (9.02 - 10.0i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 - 2.69T + 89T^{2} \) |
| 97 | \( 1 + (-5.14 + 1.09i)T + (88.6 - 39.4i)T^{2} \) |
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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
−14.26363750687556253869578606391, −13.17717943573019291901148205037, −11.41239627343201880849486208210, −10.30246179804518146014228109769, −9.454949955834465911043022859921, −8.733929355228861454679194613370, −7.54262034855222258729937280650, −6.73057491538036267485092044469, −4.26313106898376142026936397003, −1.97673196053616442700715599868,
1.68178777013014175252706167440, 3.34279302764194168010676559257, 6.17125065918329726104580600786, 7.65541470756196925021266606414, 8.614178094098597909684079934082, 9.063631025743760457678583974202, 10.38829255838508464898999639816, 11.28064280306384242704297076929, 12.68643752243459704103899238150, 13.68134755239487110061423504867
