L(s) = 1 | + (−1.27 − 0.609i)2-s + (1.73 − 0.0701i)3-s + (1.25 + 1.55i)4-s + (−0.426 − 0.155i)5-s + (−2.25 − 0.965i)6-s + (1.28 + 0.225i)7-s + (−0.656 − 2.75i)8-s + (2.99 − 0.242i)9-s + (0.449 + 0.458i)10-s + (0.00602 + 0.0165i)11-s + (2.28 + 2.60i)12-s + (1.43 + 1.71i)13-s + (−1.49 − 1.06i)14-s + (−0.749 − 0.238i)15-s + (−0.839 + 3.91i)16-s + (1.27 − 0.734i)17-s + ⋯ |
L(s) = 1 | + (−0.902 − 0.430i)2-s + (0.999 − 0.0405i)3-s + (0.628 + 0.777i)4-s + (−0.190 − 0.0694i)5-s + (−0.919 − 0.394i)6-s + (0.484 + 0.0854i)7-s + (−0.232 − 0.972i)8-s + (0.996 − 0.0809i)9-s + (0.142 + 0.144i)10-s + (0.00181 + 0.00499i)11-s + (0.659 + 0.751i)12-s + (0.398 + 0.474i)13-s + (−0.400 − 0.285i)14-s + (−0.193 − 0.0616i)15-s + (−0.209 + 0.977i)16-s + (0.308 − 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13750 - 0.230957i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13750 - 0.230957i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.27 + 0.609i)T \) |
| 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
−12.02619977166309032593346153571, −11.29053348221514614363867901724, −10.02789303817116741540816237887, −9.336161666352156215368231544373, −8.267480202000624154054338415226, −7.73621179157437631348140367349, −6.48538336537974163812787341506, −4.37793660585479103386951706014, −3.09195140549988309294886073906, −1.67172036879596234269109068133,
1.69271825619523422657578383690, 3.37473064947407627598541735745, 5.10112000725548083339382036485, 6.59792016561714381517195730585, 7.73108041729858415450956798837, 8.288080066766398105311443494463, 9.283160319927307195784785151898, 10.22000911147935513002563532602, 11.08916112853773106532057477631, 12.32638715018009114921948826361