L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + 5·13-s + (1 + 1.73i)17-s + (−1.5 + 2.59i)19-s + (2 − 1.73i)21-s + (−1 + 1.73i)23-s + (2.5 + 4.33i)25-s + 0.999·27-s + 8·29-s + (0.5 + 0.866i)31-s + (−0.999 + 1.73i)33-s + (2.5 − 4.33i)37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + 1.38·13-s + (0.242 + 0.420i)17-s + (−0.344 + 0.596i)19-s + (0.436 − 0.377i)21-s + (−0.208 + 0.361i)23-s + (0.5 + 0.866i)25-s + 0.192·27-s + 1.48·29-s + (0.0898 + 0.155i)31-s + (−0.174 + 0.301i)33-s + (0.410 − 0.711i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40438 + 0.178965i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40438 + 0.178965i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−10.83664555508901704742747250637, −9.633269741072575811828574383452, −8.495147862261865503692863698604, −8.228142734744287722552979378006, −6.89022464677755318365992258003, −5.92968103104968451139313600533, −5.43432835245337445491537850177, −3.93544302651761366684786896053, −2.68689873440682736571877829636, −1.30865545830965751972136080827,
0.956869995153323757616530987507, 2.83137774342708755548891080847, 4.15373004040385001547008461257, 4.72466592405287022646759610789, 6.05069579034749864449501904981, 6.83672102077519144216207351768, 7.916296783857286601235545036002, 8.744874791082726011338919275599, 9.790521770964003399701452324252, 10.58213763554481124016131105448