L(s) = 1 | + (−1.39 + 0.230i)2-s + (−3.03 + 1.25i)3-s + (1.89 − 0.643i)4-s + (0.551 − 1.33i)5-s + (3.94 − 2.45i)6-s + (−0.707 − 0.707i)7-s + (−2.49 + 1.33i)8-s + (5.50 − 5.50i)9-s + (−0.462 + 1.98i)10-s + (−0.508 − 0.210i)11-s + (−4.93 + 4.33i)12-s + (1.29 + 3.13i)13-s + (1.14 + 0.823i)14-s + 4.73i·15-s + (3.17 − 2.43i)16-s + 3.89i·17-s + ⋯ |
L(s) = 1 | + (−0.986 + 0.163i)2-s + (−1.75 + 0.725i)3-s + (0.946 − 0.321i)4-s + (0.246 − 0.595i)5-s + (1.61 − 1.00i)6-s + (−0.267 − 0.267i)7-s + (−0.881 + 0.471i)8-s + (1.83 − 1.83i)9-s + (−0.146 + 0.627i)10-s + (−0.153 − 0.0635i)11-s + (−1.42 + 1.25i)12-s + (0.360 + 0.869i)13-s + (0.307 + 0.220i)14-s + 1.22i·15-s + (0.793 − 0.609i)16-s + 0.944i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.390258 + 0.217038i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.390258 + 0.217038i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.230i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (3.03 - 1.25i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.551 + 1.33i)T + (-3.53 - 3.53i)T^{2} \) |
| 11 | \( 1 + (0.508 + 0.210i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-1.29 - 3.13i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 3.89iT - 17T^{2} \) |
| 19 | \( 1 + (-2.18 - 5.28i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-3.28 + 3.28i)T - 23iT^{2} \) |
| 29 | \( 1 + (5.84 - 2.42i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 + (-2.00 + 4.83i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-2.71 + 2.71i)T - 41iT^{2} \) |
| 43 | \( 1 + (-5.33 - 2.20i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 3.49iT - 47T^{2} \) |
| 53 | \( 1 + (-8.42 - 3.49i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (3.20 - 7.73i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (1.33 - 0.554i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-9.53 + 3.94i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-7.97 - 7.97i)T + 71iT^{2} \) |
| 73 | \( 1 + (9.49 - 9.49i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.36iT - 79T^{2} \) |
| 83 | \( 1 + (6.45 + 15.5i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (0.200 + 0.200i)T + 89iT^{2} \) |
| 97 | \( 1 - 6.47T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.09320098047866831267470500661, −11.15936594451774573682524109396, −10.48934705113844988999996549520, −9.688695915651262297570180749778, −8.783237582730334187977848072033, −7.19544419147975215436335534599, −6.17479855752869090142560525458, −5.45786679750590722034045566163, −4.05877797658601182146429201318, −1.16016815682476189996781967171,
0.78748888238355980547171462934, 2.67885288151935544985029575324, 5.18022829710251644046078302120, 6.22274619818210272138754310759, 6.97291898580437526752683307398, 7.81631433379077509576809174276, 9.455458889718633677983208421918, 10.39431644089131241232230481127, 11.19045503203091727308113355759, 11.70673046331558625199185015157