L(s) = 1 | + (0.5 − 0.866i)3-s + (1 + 1.73i)5-s + (0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s − 3·13-s + 1.99·15-s + (−4 + 6.92i)17-s + (0.5 + 0.866i)19-s + (−2 − 1.73i)21-s + (−4 − 6.92i)23-s + (0.500 − 0.866i)25-s − 0.999·27-s + 4·29-s + (−1.5 + 2.59i)31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.447 + 0.774i)5-s + (0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s − 0.832·13-s + 0.516·15-s + (−0.970 + 1.68i)17-s + (0.114 + 0.198i)19-s + (−0.436 − 0.377i)21-s + (−0.834 − 1.44i)23-s + (0.100 − 0.173i)25-s − 0.192·27-s + 0.742·29-s + (−0.269 + 0.466i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05425 - 0.134347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05425 - 0.134347i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + (4 - 6.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 11T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−14.34877846489454647415739956355, −13.23173806785161329894210906434, −12.29757996768351210651907455842, −10.69561359373933840087034095162, −10.12769028832827674560627736585, −8.435133764296601594801084367096, −7.24475308339824837646988399442, −6.30079322587263930132470688455, −4.27368775553655232306587359177, −2.31090490712168537397656940994,
2.59165756938551738616972537442, 4.74540548980467941915097919083, 5.71540711166325100898174515524, 7.66253600109065687526637660338, 9.081300569046940718317926347947, 9.491877347068163021800344931572, 11.14486027425762165114939820339, 12.16412271182825664677552101971, 13.36824278900699912279613894739, 14.25852321186986371167230880401