| L(s) = 1 | + (−1.56 + 0.612i)2-s + (−4.21 − 3.03i)3-s + (−3.80 + 3.52i)4-s + (5.95 − 4.05i)5-s + (8.44 + 2.15i)6-s + (8.80 − 16.2i)7-s + (9.59 − 19.9i)8-s + (8.57 + 25.6i)9-s + (−6.80 + 9.98i)10-s + (−3.97 + 26.3i)11-s + (26.7 − 3.33i)12-s + (−18.0 − 14.3i)13-s + (−3.76 + 30.8i)14-s + (−37.4 − 0.954i)15-s + (0.332 − 4.44i)16-s + (−23.7 + 7.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.551 + 0.216i)2-s + (−0.811 − 0.584i)3-s + (−0.475 + 0.441i)4-s + (0.532 − 0.363i)5-s + (0.574 + 0.146i)6-s + (0.475 − 0.879i)7-s + (0.423 − 0.880i)8-s + (0.317 + 0.948i)9-s + (−0.215 + 0.315i)10-s + (−0.109 + 0.723i)11-s + (0.643 − 0.0803i)12-s + (−0.384 − 0.306i)13-s + (−0.0719 + 0.588i)14-s + (−0.644 − 0.0164i)15-s + (0.00519 − 0.0693i)16-s + (−0.338 + 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00266005 + 0.0163622i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00266005 + 0.0163622i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.21 + 3.03i)T \) |
| 7 | \( 1 + (-8.80 + 16.2i)T \) |
| good | 2 | \( 1 + (1.56 - 0.612i)T + (5.86 - 5.44i)T^{2} \) |
| 5 | \( 1 + (-5.95 + 4.05i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (3.97 - 26.3i)T + (-1.27e3 - 392. i)T^{2} \) |
| 13 | \( 1 + (18.0 + 14.3i)T + (488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (23.7 - 7.31i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (113. + 65.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (41.5 - 134. i)T + (-1.00e4 - 6.85e3i)T^{2} \) |
| 29 | \( 1 + (28.6 + 6.53i)T + (2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + (234. - 135. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (43.1 + 40.0i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-6.52 - 3.14i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (89.1 - 42.9i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (141. + 359. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-485. - 523. i)T + (-1.11e4 + 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-178. - 121. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (133. - 144. i)T + (-1.69e4 - 2.26e5i)T^{2} \) |
| 67 | \( 1 + (111. + 193. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (909. - 207. i)T + (3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (344. + 135. i)T + (2.85e5 + 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-18.5 + 32.0i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-49.7 - 62.4i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-418. + 63.1i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 1.80e3iT - 9.12e5T^{2} \) |
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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
−13.16478399874703146320083098919, −12.16701934054119740217274346985, −10.88195776281858811954413849721, −10.00639894670618479838208306898, −8.827177978306612580650907820782, −7.58747836656521533138471056279, −6.96771069005258107425580791981, −5.33298903622330831686740526140, −4.24938522065124772149698832066, −1.63636253520696459364773126671,
0.01076953193686220196774772974, 2.07665845621962010827249957856, 4.35579945881414353751210790734, 5.54746588087158917788475768564, 6.32142938880194875395778306762, 8.331423960990245106397440038388, 9.188114914694732583126332175091, 10.20706971641093287815399317952, 10.86740319098280276686625747184, 11.81647317264377733774436345408
