L(s) = 1 | + (0.629 − 1.26i)2-s + (0.722 − 1.74i)3-s + (−1.20 − 1.59i)4-s + (−1.56 + 0.647i)5-s + (−1.75 − 2.01i)6-s + (0.707 − 0.707i)7-s + (−2.77 + 0.526i)8-s + (−0.401 − 0.401i)9-s + (−0.163 + 2.38i)10-s + (−1.93 − 4.67i)11-s + (−3.65 + 0.955i)12-s + (4.00 + 1.65i)13-s + (−0.450 − 1.34i)14-s + 3.19i·15-s + (−1.08 + 3.85i)16-s − 0.601i·17-s + ⋯ |
L(s) = 1 | + (0.445 − 0.895i)2-s + (0.417 − 1.00i)3-s + (−0.603 − 0.797i)4-s + (−0.698 + 0.289i)5-s + (−0.716 − 0.822i)6-s + (0.267 − 0.267i)7-s + (−0.982 + 0.186i)8-s + (−0.133 − 0.133i)9-s + (−0.0517 + 0.754i)10-s + (−0.584 − 1.41i)11-s + (−1.05 + 0.275i)12-s + (1.11 + 0.460i)13-s + (−0.120 − 0.358i)14-s + 0.824i·15-s + (−0.270 + 0.962i)16-s − 0.145i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.459306 - 1.38758i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.459306 - 1.38758i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.629 + 1.26i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.722 + 1.74i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (1.56 - 0.647i)T + (3.53 - 3.53i)T^{2} \) |
| 11 | \( 1 + (1.93 + 4.67i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-4.00 - 1.65i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 0.601iT - 17T^{2} \) |
| 19 | \( 1 + (-1.47 - 0.612i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.84 - 2.84i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.757 + 1.82i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 7.06T + 31T^{2} \) |
| 37 | \( 1 + (1.69 - 0.700i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (2.24 + 2.24i)T + 41iT^{2} \) |
| 43 | \( 1 + (3.46 + 8.35i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 8.14iT - 47T^{2} \) |
| 53 | \( 1 + (2.13 + 5.15i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-8.00 + 3.31i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (3.73 - 9.02i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (5.04 - 12.1i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (8.56 - 8.56i)T - 71iT^{2} \) |
| 73 | \( 1 + (-11.1 - 11.1i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.33iT - 79T^{2} \) |
| 83 | \( 1 + (0.0908 + 0.0376i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (3.11 - 3.11i)T - 89iT^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−11.71478359002903791967649447052, −11.26340556117358165599307449869, −10.26050685806070959654632450561, −8.726590043503517724000082317016, −8.041828763131454946603641434024, −6.79583025729163773522972823301, −5.53345612150327880541891919173, −3.93825601102302288961938310043, −2.86709167909356535098680999743, −1.18857991508325274173903429619,
3.19574074105838647449966550787, 4.37236470238430282663286869667, 4.99963975504838507872506254138, 6.53417791443030366802907739456, 7.81370359490072006963504627977, 8.525929276652611525344413680439, 9.514936742113650429442729360944, 10.54686143613814454637372342041, 11.93387365415323527530452598916, 12.71081892372126793789689051159