L(s) = 1 | + (−0.907 − 0.243i)2-s + (−0.315 + 1.70i)3-s + (−0.967 − 0.558i)4-s + (−2.12 + 0.695i)5-s + (0.700 − 1.46i)6-s + (−2.64 + 0.0144i)7-s + (2.07 + 2.07i)8-s + (−2.80 − 1.07i)9-s + (2.09 − 0.114i)10-s + (−0.630 − 0.363i)11-s + (1.25 − 1.47i)12-s + (−1.44 + 1.44i)13-s + (2.40 + 0.630i)14-s + (−0.515 − 3.83i)15-s + (−0.257 − 0.446i)16-s + (1.90 + 7.09i)17-s + ⋯ |
L(s) = 1 | + (−0.641 − 0.171i)2-s + (−0.182 + 0.983i)3-s + (−0.483 − 0.279i)4-s + (−0.950 + 0.311i)5-s + (0.285 − 0.599i)6-s + (−0.999 + 0.00544i)7-s + (0.732 + 0.732i)8-s + (−0.933 − 0.357i)9-s + (0.663 − 0.0363i)10-s + (−0.189 − 0.109i)11-s + (0.362 − 0.425i)12-s + (−0.400 + 0.400i)13-s + (0.642 + 0.168i)14-s + (−0.133 − 0.991i)15-s + (−0.0644 − 0.111i)16-s + (0.460 + 1.71i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0480995 + 0.226720i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0480995 + 0.226720i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.315 - 1.70i)T \) |
| 5 | \( 1 + (2.12 - 0.695i)T \) |
| 7 | \( 1 + (2.64 - 0.0144i)T \) |
good | 2 | \( 1 + (0.907 + 0.243i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (0.630 + 0.363i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.44 - 1.44i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.90 - 7.09i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.664 + 0.383i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.840 + 3.13i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + (0.209 - 0.363i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.63 - 6.08i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 4.44iT - 41T^{2} \) |
| 43 | \( 1 + (5.15 - 5.15i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.79 + 1.82i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (5.26 - 1.41i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.807 - 1.39i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.78 - 8.29i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.90 - 1.84i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 7.06iT - 71T^{2} \) |
| 73 | \( 1 + (4.08 + 15.2i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.80 + 3.35i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.83 + 1.83i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.94 - 12.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.62 - 5.62i)T + 97iT^{2} \) |
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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
−14.60297831345143648037198391508, −13.12719217628758346557776855059, −11.84904773007001102844367343294, −10.68851160962053435269015230781, −10.06515475748515033271800068235, −9.006502445152273961977772482974, −8.004078040353215822389478885290, −6.25777017293679745743225331913, −4.67767214434778840890267072660, −3.43927710989046958441187004643,
0.33149081114965854031648209193, 3.33933860931327670379555380019, 5.21528112082486383027274294198, 7.10655572312070438381036716860, 7.61065296579138581281409204746, 8.819485477302651372097285436129, 9.826477280089150003082574258138, 11.41772825595437943527039866667, 12.44786031115851504753505492986, 13.06242183593282653023884684214