L(s) = 1 | + (0.5 + 0.866i)2-s + (1.73 − i)3-s + (0.500 − 0.866i)4-s + (−1 + 2i)5-s + (1.73 + 0.999i)6-s + 3·8-s + (0.499 − 0.866i)9-s + (−2.23 + 0.133i)10-s + (1.73 − i)11-s − 2i·12-s + (0.267 + 4.46i)15-s + (0.500 + 0.866i)16-s + 0.999·18-s + (5.19 + 3i)19-s + (1.23 + 1.86i)20-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.999 − 0.577i)3-s + (0.250 − 0.433i)4-s + (−0.447 + 0.894i)5-s + (0.707 + 0.408i)6-s + 1.06·8-s + (0.166 − 0.288i)9-s + (−0.705 + 0.0423i)10-s + (0.522 − 0.301i)11-s − 0.577i·12-s + (0.0691 + 1.15i)15-s + (0.125 + 0.216i)16-s + 0.235·18-s + (1.19 + 0.688i)19-s + (0.275 + 0.417i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.79778 + 0.463228i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.79778 + 0.463228i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1 - 2i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.73 + i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.73 + i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.19 - 3i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.92 - 4i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.19 - 3i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + (1.73 + i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.73 - i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (6.92 - 4i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.25487947456240202154614958056, −9.273879107232846633437947841021, −8.223195968165444344611124371719, −7.52818824491974105645821097337, −6.92544844036748111838164293117, −6.12487502144887157331248792343, −5.02046714867910192074247657791, −3.67798413808485835004684982226, −2.77441794311190971293214826804, −1.51268520740108630265565084441,
1.46624418249582328298280527156, 2.90265449474350600316302347607, 3.61314831180320803616668573282, 4.41426330382154396575181874592, 5.31752245036304877482881594428, 7.03168043192296220180801879835, 7.70035373380780643195361024330, 8.698598722438639702841051301243, 9.191814319596625959313373766933, 10.03284032577320220672451459402