L(s) = 1 | + (0.980 − 0.195i)2-s + (0.923 − 0.382i)4-s + (−0.195 + 0.980i)5-s + (0.831 − 0.555i)8-s + i·10-s + (0.707 − 0.707i)16-s + (1.38 + 1.38i)17-s + (−1.63 + 0.324i)19-s + (0.195 + 0.980i)20-s + (1.81 − 0.750i)23-s + (−0.923 − 0.382i)25-s − 0.765·31-s + (0.555 − 0.831i)32-s + (1.63 + 1.08i)34-s + (−1.53 + 0.636i)38-s + ⋯ |
L(s) = 1 | + (0.980 − 0.195i)2-s + (0.923 − 0.382i)4-s + (−0.195 + 0.980i)5-s + (0.831 − 0.555i)8-s + i·10-s + (0.707 − 0.707i)16-s + (1.38 + 1.38i)17-s + (−1.63 + 0.324i)19-s + (0.195 + 0.980i)20-s + (1.81 − 0.750i)23-s + (−0.923 − 0.382i)25-s − 0.765·31-s + (0.555 − 0.831i)32-s + (1.63 + 1.08i)34-s + (−1.53 + 0.636i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.315495378\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.315495378\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.980 + 0.195i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.195 - 0.980i)T \) |
good | 7 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 17 | \( 1 + (-1.38 - 1.38i)T + iT^{2} \) |
| 19 | \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \) |
| 23 | \( 1 + (-1.81 + 0.750i)T + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + 0.765T + T^{2} \) |
| 37 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 47 | \( 1 + (0.785 - 0.785i)T - iT^{2} \) |
| 53 | \( 1 + (-0.636 + 0.425i)T + (0.382 - 0.923i)T^{2} \) |
| 59 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 61 | \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-1 + i)T - iT^{2} \) |
| 83 | \( 1 + (0.360 + 1.81i)T + (-0.923 + 0.382i)T^{2} \) |
| 89 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.973855102076397727992655968113, −7.987688020300533689798144594421, −7.35188426225611256983187518664, −6.43833416011933673983031243400, −6.08710028704323901273433005609, −5.07574704617367866381144978788, −4.11412218855746561682868151857, −3.44303285978182936631508708868, −2.63550261232226114014538764770, −1.58203662750875478151760848978,
1.22375330842572561179154989793, 2.49746716337463733202111632197, 3.47218195046572404938402957571, 4.29382433094226589235719264275, 5.23581189701780139755635599182, 5.41805145320786381460457940616, 6.65959780857008328759822576965, 7.29440783214380307621433689063, 8.028588935157677057687220534435, 8.846338893737845544381909487072