| L(s) = 1 | + (1.03 − 2.15i)2-s + (−1.08 − 2.25i)3-s + (−1.07 − 1.34i)4-s + (−0.904 + 0.206i)5-s − 5.99·6-s + 5.06i·7-s + (5.31 − 1.21i)8-s + (1.69 − 2.12i)9-s + (−0.493 + 2.16i)10-s + (−1.54 + 1.93i)11-s + (−1.87 + 3.88i)12-s + (2.68 + 11.7i)13-s + (10.9 + 5.25i)14-s + (1.44 + 1.81i)15-s + (4.43 − 19.4i)16-s + (−0.329 + 1.44i)17-s + ⋯ |
| L(s) = 1 | + (0.518 − 1.07i)2-s + (−0.362 − 0.752i)3-s + (−0.268 − 0.336i)4-s + (−0.180 + 0.0412i)5-s − 0.998·6-s + 0.723i·7-s + (0.664 − 0.151i)8-s + (0.188 − 0.236i)9-s + (−0.0493 + 0.216i)10-s + (−0.140 + 0.176i)11-s + (−0.155 + 0.323i)12-s + (0.206 + 0.905i)13-s + (0.779 + 0.375i)14-s + (0.0966 + 0.121i)15-s + (0.277 − 1.21i)16-s + (−0.0193 + 0.0849i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0861 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0861 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.872045 - 0.950710i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.872045 - 0.950710i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (24.7 - 35.1i)T \) |
| good | 2 | \( 1 + (-1.03 + 2.15i)T + (-2.49 - 3.12i)T^{2} \) |
| 3 | \( 1 + (1.08 + 2.25i)T + (-5.61 + 7.03i)T^{2} \) |
| 5 | \( 1 + (0.904 - 0.206i)T + (22.5 - 10.8i)T^{2} \) |
| 7 | \( 1 - 5.06iT - 49T^{2} \) |
| 11 | \( 1 + (1.54 - 1.93i)T + (-26.9 - 117. i)T^{2} \) |
| 13 | \( 1 + (-2.68 - 11.7i)T + (-152. + 73.3i)T^{2} \) |
| 17 | \( 1 + (0.329 - 1.44i)T + (-260. - 125. i)T^{2} \) |
| 19 | \( 1 + (17.4 - 13.8i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (5.29 - 6.63i)T + (-117. - 515. i)T^{2} \) |
| 29 | \( 1 + (-18.1 + 37.7i)T + (-524. - 657. i)T^{2} \) |
| 31 | \( 1 + (12.6 + 6.10i)T + (599. + 751. i)T^{2} \) |
| 37 | \( 1 + 47.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-54.7 - 26.3i)T + (1.04e3 + 1.31e3i)T^{2} \) |
| 47 | \( 1 + (14.3 + 18.0i)T + (-491. + 2.15e3i)T^{2} \) |
| 53 | \( 1 + (13.6 - 59.7i)T + (-2.53e3 - 1.21e3i)T^{2} \) |
| 59 | \( 1 + (-4.12 + 18.0i)T + (-3.13e3 - 1.51e3i)T^{2} \) |
| 61 | \( 1 + (-10.4 - 21.6i)T + (-2.32e3 + 2.90e3i)T^{2} \) |
| 67 | \( 1 + (79.2 + 99.4i)T + (-998. + 4.37e3i)T^{2} \) |
| 71 | \( 1 + (-90.0 + 71.8i)T + (1.12e3 - 4.91e3i)T^{2} \) |
| 73 | \( 1 + (-70.3 + 16.0i)T + (4.80e3 - 2.31e3i)T^{2} \) |
| 79 | \( 1 - 47.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + (72.7 - 35.0i)T + (4.29e3 - 5.38e3i)T^{2} \) |
| 89 | \( 1 + (49.6 + 103. i)T + (-4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (94.6 - 118. i)T + (-2.09e3 - 9.17e3i)T^{2} \) |
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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
−15.29090480054261575933152350132, −13.81994280409560666236932376299, −12.69895135234274273574654284988, −12.03878064325744866221882744497, −11.13486983661223771688128723123, −9.580195460788478254661270622884, −7.71220563754594643833767102148, −6.14876434921304679947247581020, −4.08421578942237215281156856043, −1.99614283196524447820319288315,
4.21208420657983123830988124826, 5.39326931541779733280061239440, 6.90474821268282362926798290676, 8.209317010839788677889416475357, 10.23569362276981858399455617957, 10.97745189875507956684578060691, 12.93427757237604940991636888605, 13.96976786985994499236700738485, 15.18957308517891317758899143480, 15.92839816355798580819085266395
