L(s) = 1 | + (−0.923 + 0.382i)3-s + (1.30 + 1.30i)7-s + (0.707 − 0.707i)9-s + (0.292 + 0.707i)13-s + (0.541 + 1.30i)19-s + (−1.70 − 0.707i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + 0.765·31-s + (0.707 − 1.70i)37-s + (−0.541 − 0.541i)39-s + (−1.30 − 0.541i)43-s + 2.41i·49-s + (−1 − 0.999i)57-s + (0.707 − 0.292i)61-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)3-s + (1.30 + 1.30i)7-s + (0.707 − 0.707i)9-s + (0.292 + 0.707i)13-s + (0.541 + 1.30i)19-s + (−1.70 − 0.707i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + 0.765·31-s + (0.707 − 1.70i)37-s + (−0.541 − 0.541i)39-s + (−1.30 − 0.541i)43-s + 2.41i·49-s + (−1 − 0.999i)57-s + (0.707 − 0.292i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.081974895\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.081974895\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.923 - 0.382i)T \) |
good | 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - 0.765T + T^{2} \) |
| 37 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 - 0.765iT - T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−9.060051334885394777286508586732, −8.329068215652048172350978290375, −7.65787612362301787721560841354, −6.58309021507933079807992295538, −5.79835074883631151497514687103, −5.37515121913587532109360754552, −4.50829667785073276366896058865, −3.77517641070543661383595041347, −2.31842780243654169438872265015, −1.43064534959063924852575815594,
0.860012202235964090556655836363, 1.66989907582111133712343168679, 3.12299953769093333479382334896, 4.36283664209307485351491120774, 4.82849559737341741729958851297, 5.58387979466131251070812516145, 6.58651204656081720602391047158, 7.22510282726203949048534033816, 7.86593124119516729669967765815, 8.409960443717712875088211715039