L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (−0.126 + 2.23i)5-s + i·6-s + (−2.47 + 0.942i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (2.18 − 0.455i)10-s + (1.55 + 2.69i)11-s + (0.965 − 0.258i)12-s + (−3.40 + 3.40i)13-s + (1.55 + 2.14i)14-s + (0.700 − 2.12i)15-s + (0.500 − 0.866i)16-s + (−1.37 + 5.14i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (−0.0566 + 0.998i)5-s + 0.408i·6-s + (−0.934 + 0.356i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (0.692 − 0.144i)10-s + (0.468 + 0.811i)11-s + (0.278 − 0.0747i)12-s + (−0.945 + 0.945i)13-s + (0.414 + 0.572i)14-s + (0.180 − 0.548i)15-s + (0.125 − 0.216i)16-s + (−0.334 + 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.503643 + 0.343880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.503643 + 0.343880i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (0.126 - 2.23i)T \) |
| 7 | \( 1 + (2.47 - 0.942i)T \) |
good | 11 | \( 1 + (-1.55 - 2.69i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.40 - 3.40i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.37 - 5.14i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.61 + 6.26i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.08 + 1.36i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 4.49iT - 29T^{2} \) |
| 31 | \( 1 + (7.98 - 4.61i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.929 + 3.47i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.51iT - 41T^{2} \) |
| 43 | \( 1 + (3.86 + 3.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.94 - 1.05i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.890 + 3.32i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.666 + 1.15i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.8 - 6.27i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.43 - 1.72i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 9.22T + 71T^{2} \) |
| 73 | \( 1 + (-7.95 - 2.13i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (1.65 + 0.957i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.97 + 8.97i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.03 - 3.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.69 - 2.69i)T + 97iT^{2} \) |
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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
−12.42971285679885643142828332902, −11.52679281590533596445718926255, −10.72670586078066780252928007559, −9.749488173492858774255356758023, −8.995181282170582024942965209140, −7.09942977934142549402835691025, −6.73181695911473184163817483154, −5.07196330486539237125772314292, −3.59519762217931163150889183482, −2.20397868591573535635903639929,
0.58981192959008811344652750719, 3.58629041212724973734336450085, 5.05961361567717802274034798878, 5.82076631055171216947327059339, 7.08186063875238493355963375904, 8.077444922530513671007547724257, 9.430552527016304883980717687713, 9.818923631441294393061345335659, 11.26583355352650341225883344923, 12.31735722620473101999084741896