L(s) = 1 | + (−0.707 + 0.707i)3-s + 1.41i·7-s − 1.00i·9-s + (−1 + i)13-s + 6·17-s + (−4.24 + 4.24i)19-s + (−1.00 − 1.00i)21-s + 8.48i·23-s − 5i·25-s + (0.707 + 0.707i)27-s + (−6 + 6i)29-s − 1.41·31-s + (−5 − 5i)37-s − 1.41i·39-s + 6i·41-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + 0.534i·7-s − 0.333i·9-s + (−0.277 + 0.277i)13-s + 1.45·17-s + (−0.973 + 0.973i)19-s + (−0.218 − 0.218i)21-s + 1.76i·23-s − i·25-s + (0.136 + 0.136i)27-s + (−1.11 + 1.11i)29-s − 0.254·31-s + (−0.821 − 0.821i)37-s − 0.226i·39-s + 0.937i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.561004 + 0.839602i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.561004 + 0.839602i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
good | 5 | \( 1 + 5iT^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + (4.24 - 4.24i)T - 19iT^{2} \) |
| 23 | \( 1 - 8.48iT - 23T^{2} \) |
| 29 | \( 1 + (6 - 6i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 - 4.24i)T + 43iT^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + (-6 - 6i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (5 - 5i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.82 - 2.82i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−10.52110694422570772889004335844, −9.782601191833913671875059868283, −9.036757784577249241503103294656, −8.020857896992264591328525490687, −7.14773171218612921671672005667, −5.89082707323611921294367441610, −5.44579491230146236877156114193, −4.19071049242225518080767056514, −3.20942371647458972517212775952, −1.65570692786830293621000380784,
0.54199357997023415177221832529, 2.13967858469280935062734473594, 3.52307585605867997350083267035, 4.70005743981252553254052781398, 5.62739466846925634789877862234, 6.63279219802025869504953037582, 7.39194044083058310715094072917, 8.209554747461376406928244404405, 9.222051040982898383324703723179, 10.27406048793311052029188765143