L(s) = 1 | + (1.24 − 0.664i)2-s + (0.517 + 1.65i)3-s + (1.11 − 1.65i)4-s + (−0.521 + 1.94i)5-s + (1.74 + 1.71i)6-s + (−0.322 + 0.558i)7-s + (0.290 − 2.81i)8-s + (−2.46 + 1.71i)9-s + (0.642 + 2.77i)10-s + (−1.49 − 5.59i)11-s + (3.32 + 0.986i)12-s + (0.530 − 1.97i)13-s + (−0.0311 + 0.910i)14-s + (−3.48 + 0.144i)15-s + (−1.50 − 3.70i)16-s + 3.05i·17-s + ⋯ |
L(s) = 1 | + (0.882 − 0.470i)2-s + (0.298 + 0.954i)3-s + (0.558 − 0.829i)4-s + (−0.233 + 0.869i)5-s + (0.712 + 0.701i)6-s + (−0.121 + 0.210i)7-s + (0.102 − 0.994i)8-s + (−0.821 + 0.570i)9-s + (0.203 + 0.877i)10-s + (−0.451 − 1.68i)11-s + (0.958 + 0.284i)12-s + (0.147 − 0.548i)13-s + (−0.00833 + 0.243i)14-s + (−0.899 + 0.0373i)15-s + (−0.377 − 0.926i)16-s + 0.741i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75348 + 0.144361i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75348 + 0.144361i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.24 + 0.664i)T \) |
| 3 | \( 1 + (-0.517 - 1.65i)T \) |
good | 5 | \( 1 + (0.521 - 1.94i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (0.322 - 0.558i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.49 + 5.59i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.530 + 1.97i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 - 3.05iT - 17T^{2} \) |
| 19 | \( 1 + (-4.11 + 4.11i)T - 19iT^{2} \) |
| 23 | \( 1 + (7.05 - 4.07i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.20 - 4.49i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (0.147 - 0.0854i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.65 - 2.65i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.983 + 1.70i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.821 + 0.220i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (4.02 - 6.97i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.25 - 3.25i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.506 - 0.135i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.00 + 0.805i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (2.17 + 0.583i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 + 9.33iT - 73T^{2} \) |
| 79 | \( 1 + (-9.44 - 5.45i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.3 + 3.31i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 3.86T + 89T^{2} \) |
| 97 | \( 1 + (7.33 - 12.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.46465494868199436608834734604, −11.93535909874819844818001440983, −10.91739593114521456361662821250, −10.56044070911277543371345197280, −9.253279717074005223573062028704, −7.87425241669611631243059768271, −6.18597548042749552213525639282, −5.24914668080165457660353331595, −3.58604206715700182600149356328, −2.94124570498055163400936717707,
2.17385203792850702732016033906, 4.06276306051349242329185740427, 5.29393097860765272348012110655, 6.69955067484367312933139707781, 7.59064288612886943685827110605, 8.456566365193760719174438878905, 9.894627587011534139467563320226, 11.84973499357915787511171492463, 12.20637707162695909557612313858, 13.08023718879704430769617363133