| L(s) = 1 | + (0.228 + 1.71i)3-s − 1.84·5-s + (0.286 + 2.63i)7-s + (−2.89 + 0.783i)9-s − i·11-s + 6.65i·13-s + (−0.420 − 3.16i)15-s + 6.91·17-s + 0.932i·19-s + (−4.45 + 1.09i)21-s + 6.36i·23-s − 1.60·25-s + (−2.00 − 4.79i)27-s − 1.98i·29-s − 5.52i·31-s + ⋯ |
| L(s) = 1 | + (0.131 + 0.991i)3-s − 0.823·5-s + (0.108 + 0.994i)7-s + (−0.965 + 0.261i)9-s − 0.301i·11-s + 1.84i·13-s + (−0.108 − 0.816i)15-s + 1.67·17-s + 0.214i·19-s + (−0.971 + 0.238i)21-s + 1.32i·23-s − 0.321·25-s + (−0.386 − 0.922i)27-s − 0.367i·29-s − 0.991i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9691742254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9691742254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.228 - 1.71i)T \) |
| 7 | \( 1 + (-0.286 - 2.63i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.84T + 5T^{2} \) |
| 13 | \( 1 - 6.65iT - 13T^{2} \) |
| 17 | \( 1 - 6.91T + 17T^{2} \) |
| 19 | \( 1 - 0.932iT - 19T^{2} \) |
| 23 | \( 1 - 6.36iT - 23T^{2} \) |
| 29 | \( 1 + 1.98iT - 29T^{2} \) |
| 31 | \( 1 + 5.52iT - 31T^{2} \) |
| 37 | \( 1 - 0.344T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 6.02iT - 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 7.35T + 67T^{2} \) |
| 71 | \( 1 + 7.58iT - 71T^{2} \) |
| 73 | \( 1 + 7.92iT - 73T^{2} \) |
| 79 | \( 1 + 0.993T + 79T^{2} \) |
| 83 | \( 1 - 6.57T + 83T^{2} \) |
| 89 | \( 1 - 9.35T + 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−9.434257943020635422062026181742, −9.135218292535247301596915051938, −8.088066648983126894595032660731, −7.63173347960792641144581553099, −6.27014210418083110926405309188, −5.58072596641391051945105749015, −4.67802102794921281747884109121, −3.83187289189420725940878192817, −3.14188576471660863170162575147, −1.83047509793120803619091460990,
0.37255681701387092805509811717, 1.32606865091174793641202262814, 2.95432535883596426529061011836, 3.50647090830974774485080246231, 4.75419304307904006748781676808, 5.65632661636915710891392207551, 6.64854693750720877363316247981, 7.44022925596952677741382471551, 7.966792657651332361092957148613, 8.341943673943804820701650976820
