L(s) = 1 | + (2.18 + 2.18i)3-s + (2.18 + 0.456i)5-s + (−1.79 + 1.79i)7-s + 6.58i·9-s + 0.913i·11-s + (1.73 − 1.73i)13-s + (3.79 + 5.79i)15-s + (3 + 3i)17-s − 3.46·19-s − 7.84·21-s + (−3.79 − 3.79i)23-s + (4.58 + 1.99i)25-s + (−7.84 + 7.84i)27-s − 5.29i·29-s − 7.58i·31-s + ⋯ |
L(s) = 1 | + (1.26 + 1.26i)3-s + (0.978 + 0.204i)5-s + (−0.677 + 0.677i)7-s + 2.19i·9-s + 0.275i·11-s + (0.480 − 0.480i)13-s + (0.978 + 1.49i)15-s + (0.727 + 0.727i)17-s − 0.794·19-s − 1.71·21-s + (−0.790 − 0.790i)23-s + (0.916 + 0.399i)25-s + (−1.50 + 1.50i)27-s − 0.982i·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.798456375\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.798456375\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.18 - 0.456i)T \) |
good | 3 | \( 1 + (-2.18 - 2.18i)T + 3iT^{2} \) |
| 7 | \( 1 + (1.79 - 1.79i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.913iT - 11T^{2} \) |
| 13 | \( 1 + (-1.73 + 1.73i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3 - 3i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + (3.79 + 3.79i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.29iT - 29T^{2} \) |
| 31 | \( 1 + 7.58iT - 31T^{2} \) |
| 37 | \( 1 + (-5.19 - 5.19i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 + (0.361 + 0.361i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.79 - 3.79i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.64 + 2.64i)T - 53iT^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 + 6.20T + 61T^{2} \) |
| 67 | \( 1 + (-0.361 + 0.361i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.41iT - 71T^{2} \) |
| 73 | \( 1 + (-8.58 + 8.58i)T - 73iT^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (-3.10 - 3.10i)T + 83iT^{2} \) |
| 89 | \( 1 + 15.1iT - 89T^{2} \) |
| 97 | \( 1 + (8.58 + 8.58i)T + 97iT^{2} \) |
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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
−9.856448261194888503710062972655, −9.299745861185072064603111908585, −8.433254259832011530356671681664, −7.908047491852167191251347181025, −6.34135469409504657404182151176, −5.79297067414972358870655135363, −4.62662554122808858753814277222, −3.72632299089845978675905314143, −2.79179106005629997867376090643, −2.09440825238449767662747074273,
1.04426950058689916098574537510, 1.98588428139939954492154962672, 3.03483134350125113680047390474, 3.84021470919061395016306762876, 5.40802673561009833059074704358, 6.49379770244957405055843620869, 6.88041619873315343301080616770, 7.80975842191694806694419250983, 8.628099872639163101763114235118, 9.281504774037207761258424687010