| L(s) = 1 | + (−2.02 − 8.86i)2-s + (20.8 + 19.3i)3-s + (−45.7 + 22.0i)4-s + (−27.6 + 4.16i)5-s + (129. − 224. i)6-s + (100. + 174. i)7-s + (106. + 133. i)8-s + (42.5 + 567. i)9-s + (92.8 + 236. i)10-s + (404. + 194. i)11-s + (−1.38e3 − 426. i)12-s + (254. − 649. i)13-s + (1.34e3 − 1.24e3i)14-s + (−657. − 448. i)15-s + (−44.9 + 56.4i)16-s + (−836. − 126. i)17-s + ⋯ |
| L(s) = 1 | + (−0.357 − 1.56i)2-s + (1.34 + 1.24i)3-s + (−1.42 + 0.688i)4-s + (−0.493 + 0.0744i)5-s + (1.47 − 2.54i)6-s + (0.778 + 1.34i)7-s + (0.587 + 0.736i)8-s + (0.175 + 2.33i)9-s + (0.293 + 0.747i)10-s + (1.00 + 0.485i)11-s + (−2.77 − 0.854i)12-s + (0.418 − 1.06i)13-s + (1.83 − 1.70i)14-s + (−0.754 − 0.514i)15-s + (−0.0439 + 0.0551i)16-s + (−0.701 − 0.105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0745i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.90696 - 0.0712248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90696 - 0.0712248i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (534. + 1.21e4i)T \) |
| good | 2 | \( 1 + (2.02 + 8.86i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (-20.8 - 19.3i)T + (18.1 + 242. i)T^{2} \) |
| 5 | \( 1 + (27.6 - 4.16i)T + (2.98e3 - 921. i)T^{2} \) |
| 7 | \( 1 + (-100. - 174. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-404. - 194. i)T + (1.00e5 + 1.25e5i)T^{2} \) |
| 13 | \( 1 + (-254. + 649. i)T + (-2.72e5 - 2.52e5i)T^{2} \) |
| 17 | \( 1 + (836. + 126. i)T + (1.35e6 + 4.18e5i)T^{2} \) |
| 19 | \( 1 + (3.05 - 40.7i)T + (-2.44e6 - 3.69e5i)T^{2} \) |
| 23 | \( 1 + (-2.37e3 + 1.61e3i)T + (2.35e6 - 5.99e6i)T^{2} \) |
| 29 | \( 1 + (-4.37e3 + 4.05e3i)T + (1.53e6 - 2.04e7i)T^{2} \) |
| 31 | \( 1 + (459. + 141. i)T + (2.36e7 + 1.61e7i)T^{2} \) |
| 37 | \( 1 + (7.17e3 - 1.24e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (1.86e3 + 8.18e3i)T + (-1.04e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-1.83e3 + 881. i)T + (1.42e8 - 1.79e8i)T^{2} \) |
| 53 | \( 1 + (-2.49e3 - 6.36e3i)T + (-3.06e8 + 2.84e8i)T^{2} \) |
| 59 | \( 1 + (-1.62e4 + 2.04e4i)T + (-1.59e8 - 6.96e8i)T^{2} \) |
| 61 | \( 1 + (-8.10e3 + 2.50e3i)T + (6.97e8 - 4.75e8i)T^{2} \) |
| 67 | \( 1 + (-3.01e3 + 4.02e4i)T + (-1.33e9 - 2.01e8i)T^{2} \) |
| 71 | \( 1 + (2.23e4 + 1.52e4i)T + (6.59e8 + 1.67e9i)T^{2} \) |
| 73 | \( 1 + (-8.58e3 + 2.18e4i)T + (-1.51e9 - 1.41e9i)T^{2} \) |
| 79 | \( 1 + (2.63e4 + 4.55e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-5.46e4 - 5.07e4i)T + (2.94e8 + 3.92e9i)T^{2} \) |
| 89 | \( 1 + (3.04e4 + 2.82e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + (1.00e5 + 4.85e4i)T + (5.35e9 + 6.71e9i)T^{2} \) |
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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
−15.16251762055227245004091676991, −13.72205309662522228732306122400, −12.20799639122971928686865915433, −11.16210880011076761521582824004, −10.03736543571065434535942382671, −8.917675721856014038988110204416, −8.370827269683757125839451603457, −4.65747224853519417765619020794, −3.36445299477615402638606067216, −2.15204891431992511095949777685,
1.18358959377671337788422188864, 4.04000340606052534260153800323, 6.67522689996870596388395556330, 7.28563259671460886951825337125, 8.337334283890907922079578495760, 9.095433147417450553699855226768, 11.53072958896854207427099880459, 13.37465283538982277214070567710, 14.14798002026929748490717590110, 14.63722334942363008940635409237
