L(s) = 1 | − 3-s + (−0.456 + 2.18i)5-s + 4.37i·7-s + 9-s + 5.58i·11-s − 4.37·13-s + (0.456 − 2.18i)15-s − 5.58i·17-s − 4i·19-s − 4.37i·21-s + (−4.58 − 1.99i)25-s − 27-s − 2.55i·29-s + 5.29·31-s − 5.58i·33-s + ⋯ |
L(s) = 1 | − 0.577·3-s + (−0.204 + 0.978i)5-s + 1.65i·7-s + 0.333·9-s + 1.68i·11-s − 1.21·13-s + (0.117 − 0.565i)15-s − 1.35i·17-s − 0.917i·19-s − 0.955i·21-s + (−0.916 − 0.399i)25-s − 0.192·27-s − 0.473i·29-s + 0.950·31-s − 0.971i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0180398 - 0.643641i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0180398 - 0.643641i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (0.456 - 2.18i)T \) |
good | 7 | \( 1 - 4.37iT - 7T^{2} \) |
| 11 | \( 1 - 5.58iT - 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 + 5.58iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2.55iT - 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 + 2.55T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 7.84T + 53T^{2} \) |
| 59 | \( 1 - 1.58iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 - 7.16T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
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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
−10.30508222830970727223247525740, −9.711506307625186955045997395140, −9.019650052226107325568805337968, −7.63483157302954992395608685344, −7.08183763562744260835074605427, −6.25246104090554306259571919337, −5.14422831748214880773073065215, −4.57507824947107299145903836815, −2.77687933069760269909593992223, −2.26572542118183947364170604892,
0.32825710752831850795152923753, 1.42246539150618392954900810335, 3.50492594042962787816889581019, 4.22841036295965053707787847387, 5.19838086612090823348027642422, 6.08923295895327205390420879592, 7.07259584106963213847970514676, 8.020583497120243485931625636742, 8.554757046386821466209480939839, 9.882300261679140257290872798958