| L(s) = 1 | + (0.821 − 1.15i)2-s + (1.73 + 0.0701i)3-s + (−0.649 − 1.89i)4-s + (0.426 − 0.155i)5-s + (1.50 − 1.93i)6-s + (−1.28 + 0.225i)7-s + (−2.71 − 0.807i)8-s + (2.99 + 0.242i)9-s + (0.171 − 0.618i)10-s + (0.00602 − 0.0165i)11-s + (−0.990 − 3.31i)12-s + (−1.43 + 1.71i)13-s + (−0.793 + 1.66i)14-s + (0.749 − 0.238i)15-s + (−3.15 + 2.45i)16-s + (1.27 + 0.734i)17-s + ⋯ |
| L(s) = 1 | + (0.581 − 0.813i)2-s + (0.999 + 0.0405i)3-s + (−0.324 − 0.945i)4-s + (0.190 − 0.0694i)5-s + (0.613 − 0.789i)6-s + (−0.484 + 0.0854i)7-s + (−0.958 − 0.285i)8-s + (0.996 + 0.0809i)9-s + (0.0543 − 0.195i)10-s + (0.00181 − 0.00499i)11-s + (−0.286 − 0.958i)12-s + (−0.398 + 0.474i)13-s + (−0.211 + 0.443i)14-s + (0.193 − 0.0616i)15-s + (−0.789 + 0.614i)16-s + (0.308 + 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63607 - 1.15018i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63607 - 1.15018i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.821 + 1.15i)T \) |
| 3 | \( 1 + (-1.73 - 0.0701i)T \) |
| good | 5 | \( 1 + (-0.426 + 0.155i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.28 - 0.225i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.00602 + 0.0165i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.43 - 1.71i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.27 - 0.734i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.677 - 1.17i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.369 - 2.09i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.56 + 4.67i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (8.87 + 1.56i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.58 - 2.64i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.56 + 1.86i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (10.1 + 3.71i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.791 - 4.48i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + (3.75 + 10.3i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-8.87 + 1.56i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.70 - 3.10i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (5.03 - 8.72i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.339 + 0.587i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.19 + 6.19i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.29 - 1.53i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.103 - 0.0598i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.4 + 3.81i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.31505697791795762047292898491, −11.27515634274707439744216357373, −9.898039802523683247206725135707, −9.601627190104834664232010006020, −8.392397549479710193454860599728, −7.01269782960559767956042241022, −5.65895418004332531736670328731, −4.25274910000894367755671482037, −3.19888809794751246755119065260, −1.88492126843989388520310866738,
2.69393366403258973188999990618, 3.80879315743677042086693565306, 5.14071439137778762519505325343, 6.52366144850818761902653398812, 7.42515931725830454442368854969, 8.372369982568431320049452298059, 9.333173274561899341898254832094, 10.30414558266341159839157096558, 11.97408875380128504788988921518, 12.87827080767831139269462198948
