L(s) = 1 | + (1.35 − 0.388i)2-s + (1.03 − 1.03i)3-s + (1.69 − 1.05i)4-s + (1.01 − 1.81i)6-s + 1.49i·7-s + (1.89 − 2.09i)8-s + 0.836i·9-s + (0.423 + 0.423i)11-s + (0.666 − 2.86i)12-s + (1.85 − 1.85i)13-s + (0.581 + 2.03i)14-s + (1.76 − 3.58i)16-s − 6.50·17-s + (0.325 + 1.13i)18-s + (−1.75 + 1.75i)19-s + ⋯ |
L(s) = 1 | + (0.961 − 0.274i)2-s + (0.600 − 0.600i)3-s + (0.849 − 0.528i)4-s + (0.412 − 0.742i)6-s + 0.565i·7-s + (0.671 − 0.741i)8-s + 0.278i·9-s + (0.127 + 0.127i)11-s + (0.192 − 0.827i)12-s + (0.515 − 0.515i)13-s + (0.155 + 0.543i)14-s + (0.441 − 0.897i)16-s − 1.57·17-s + (0.0766 + 0.268i)18-s + (−0.403 + 0.403i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.58960 - 1.17230i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.58960 - 1.17230i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.35 + 0.388i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.03 + 1.03i)T - 3iT^{2} \) |
| 7 | \( 1 - 1.49iT - 7T^{2} \) |
| 11 | \( 1 + (-0.423 - 0.423i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.85 + 1.85i)T - 13iT^{2} \) |
| 17 | \( 1 + 6.50T + 17T^{2} \) |
| 19 | \( 1 + (1.75 - 1.75i)T - 19iT^{2} \) |
| 23 | \( 1 + 7.19iT - 23T^{2} \) |
| 29 | \( 1 + (6.57 - 6.57i)T - 29iT^{2} \) |
| 31 | \( 1 + 6.75T + 31T^{2} \) |
| 37 | \( 1 + (1.95 + 1.95i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.70iT - 41T^{2} \) |
| 43 | \( 1 + (-6.13 - 6.13i)T + 43iT^{2} \) |
| 47 | \( 1 - 6.65T + 47T^{2} \) |
| 53 | \( 1 + (5.29 + 5.29i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.91 - 5.91i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.43 - 1.43i)T - 61iT^{2} \) |
| 67 | \( 1 + (-6.35 + 6.35i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.08iT - 71T^{2} \) |
| 73 | \( 1 - 2.43iT - 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + (2.81 - 2.81i)T - 83iT^{2} \) |
| 89 | \( 1 + 10.5iT - 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−11.07470874969153551229728955904, −10.71933953706985857698845953814, −9.213104358878180774238401926695, −8.342899990056513254321613486085, −7.23453188359593186353590364411, −6.35241833578953384390376307886, −5.27732677660795652798243653522, −4.10708093743519607881019862792, −2.75729115148102497889878853465, −1.84707809157056461338657992796,
2.22126381900086882861725382774, 3.80020830423017680902026643815, 4.08487245027669400888291316243, 5.52902822008514472315183007253, 6.63831360199254143802552145738, 7.46270097848128472412369246589, 8.724395189561959264800272486080, 9.416198611734468631687995042234, 10.79028033855448701381557233958, 11.31882962091781004333763105572