| L(s) = 1 | + (−0.950 + 1.04i)2-s + (0.853 + 2.62i)3-s + (−0.191 − 1.99i)4-s + (2.18 − 0.485i)5-s + (−3.56 − 1.60i)6-s + 2.67i·7-s + (2.26 + 1.69i)8-s + (−3.74 + 2.72i)9-s + (−1.56 + 2.74i)10-s + (0.305 − 0.420i)11-s + (5.06 − 2.20i)12-s + (4.26 − 3.09i)13-s + (−2.79 − 2.54i)14-s + (3.13 + 5.32i)15-s + (−3.92 + 0.764i)16-s + (−6.80 − 2.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.740i)2-s + (0.492 + 1.51i)3-s + (−0.0959 − 0.995i)4-s + (0.976 − 0.217i)5-s + (−1.45 − 0.655i)6-s + 1.01i·7-s + (0.801 + 0.598i)8-s + (−1.24 + 0.907i)9-s + (−0.495 + 0.868i)10-s + (0.0921 − 0.126i)11-s + (1.46 − 0.636i)12-s + (1.18 − 0.859i)13-s + (−0.748 − 0.679i)14-s + (0.810 + 1.37i)15-s + (−0.981 + 0.191i)16-s + (−1.65 − 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.517245 + 0.996890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.517245 + 0.996890i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.950 - 1.04i)T \) |
| 5 | \( 1 + (-2.18 + 0.485i)T \) |
| good | 3 | \( 1 + (-0.853 - 2.62i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 2.67iT - 7T^{2} \) |
| 11 | \( 1 + (-0.305 + 0.420i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-4.26 + 3.09i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (6.80 + 2.21i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (3.74 + 1.21i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.11 - 2.91i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.63 + 1.18i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.54 + 7.82i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.62 - 1.17i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (5.16 - 3.75i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.41T + 43T^{2} \) |
| 47 | \( 1 + (-0.748 + 0.243i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.71 - 5.28i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.67 - 5.05i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.51 + 6.21i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.98 + 6.09i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.885 - 2.72i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.839 - 1.15i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.10 + 6.49i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.170 + 0.524i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (11.0 + 7.99i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.85 + 1.57i)T + (78.4 - 57.0i)T^{2} \) |
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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
−13.23000538551470847226541247023, −11.29635922350107634801439244866, −10.46933929034162392917954594869, −9.598362169617487218799586046755, −8.871052985620962769239491563226, −8.383217772526940698830529348052, −6.34928041654889315701887373481, −5.53594315950584905957653528002, −4.43230743851892240026558499439, −2.46918846337047850724995058587,
1.39031972381609015406615167407, 2.34701284521182375771037161868, 4.04282808369749002037444811669, 6.64231252830762113905768042387, 6.80510421016051094044516659653, 8.379408653267153974291103449315, 8.861940949464948377790363861890, 10.33041880084799147297631056737, 11.00679268921961044383490161507, 12.27706242850796621852133730708
