| L(s) = 1 | + (0.380 − 0.117i)2-s + (0.365 + 0.930i)3-s + (−1.52 + 1.03i)4-s + (0.996 + 0.150i)5-s + (0.248 + 0.310i)6-s + (2.64 + 0.00303i)7-s + (−0.952 + 1.19i)8-s + (−0.733 + 0.680i)9-s + (0.396 − 0.0597i)10-s + (1.00 + 0.928i)11-s + (−1.52 − 1.03i)12-s + (−1.26 + 5.55i)13-s + (1.00 − 0.309i)14-s + (0.224 + 0.982i)15-s + (1.12 − 2.86i)16-s + (0.594 − 7.92i)17-s + ⋯ |
| L(s) = 1 | + (0.268 − 0.0829i)2-s + (0.210 + 0.537i)3-s + (−0.760 + 0.518i)4-s + (0.445 + 0.0671i)5-s + (0.101 + 0.126i)6-s + (0.999 + 0.00114i)7-s + (−0.336 + 0.422i)8-s + (−0.244 + 0.226i)9-s + (0.125 − 0.0188i)10-s + (0.301 + 0.280i)11-s + (−0.439 − 0.299i)12-s + (−0.351 + 1.54i)13-s + (0.268 − 0.0825i)14-s + (0.0579 + 0.253i)15-s + (0.280 − 0.715i)16-s + (0.144 − 1.92i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16742 + 0.530158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16742 + 0.530158i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.365 - 0.930i)T \) |
| 7 | \( 1 + (-2.64 - 0.00303i)T \) |
| good | 2 | \( 1 + (-0.380 + 0.117i)T + (1.65 - 1.12i)T^{2} \) |
| 5 | \( 1 + (-0.996 - 0.150i)T + (4.77 + 1.47i)T^{2} \) |
| 11 | \( 1 + (-1.00 - 0.928i)T + (0.822 + 10.9i)T^{2} \) |
| 13 | \( 1 + (1.26 - 5.55i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.594 + 7.92i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (1.97 + 3.42i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.416 + 5.56i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (-8.45 + 4.07i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (2.53 - 4.39i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.20 - 2.86i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (4.23 - 5.30i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (0.406 + 0.509i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (10.4 - 3.22i)T + (38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (-6.20 + 4.22i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (3.22 - 0.486i)T + (56.3 - 17.3i)T^{2} \) |
| 61 | \( 1 + (0.0921 + 0.0628i)T + (22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (2.03 - 3.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.85 - 0.892i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-6.83 - 2.10i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (2.75 + 4.76i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.325 + 1.42i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (4.54 - 4.22i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 - 4.96T + 97T^{2} \) |
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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
−13.51865216533010453776344974812, −12.01022614842928168066270147159, −11.45771150635417334418519126299, −9.902735115079473284832868327646, −9.129723038585441160226883408127, −8.196310067497641817575456728399, −6.78272362645080207336404211051, −4.88440133144154065164580573534, −4.44634864396080172475325808372, −2.56125375983933906142116189208,
1.55286223038874426748106799200, 3.78129877584304060719787260894, 5.35465101668680401052723828914, 6.08182747355134603560016692849, 7.889121393496211006890239184709, 8.552475271785307594248068432640, 9.921964070807379369216093765615, 10.76219188288876236184921477277, 12.24719972685824490530822151442, 13.05300660309377524635452201717
