L(s) = 1 | + 1.52i·2-s + (−15.4 + 2.09i)3-s + 29.6·4-s − 32.5·5-s + (−3.19 − 23.4i)6-s + 93.8i·8-s + (234. − 64.8i)9-s − 49.4i·10-s + 321. i·11-s + (−458. + 62.3i)12-s − 609. i·13-s + (502. − 68.2i)15-s + 807.·16-s − 31.0·17-s + (98.6 + 356. i)18-s + 1.31e3i·19-s + ⋯ |
L(s) = 1 | + 0.268i·2-s + (−0.990 + 0.134i)3-s + 0.927·4-s − 0.581·5-s + (−0.0362 − 0.266i)6-s + 0.518i·8-s + (0.963 − 0.266i)9-s − 0.156i·10-s + 0.801i·11-s + (−0.919 + 0.124i)12-s − 0.999i·13-s + (0.576 − 0.0783i)15-s + 0.788·16-s − 0.0260·17-s + (0.0717 + 0.259i)18-s + 0.836i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.6519058672\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6519058672\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (15.4 - 2.09i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.52iT - 32T^{2} \) |
| 5 | \( 1 + 32.5T + 3.12e3T^{2} \) |
| 11 | \( 1 - 321. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 609. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 31.0T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.31e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.06e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 8.10e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.85e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 9.76e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.96e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.70e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.82e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.31e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 6.67e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.50e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.14e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 3.29e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 5.14e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.32e5iT - 8.58e9T^{2} \) |
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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
−12.05126917995280165933096841590, −11.87519102275664507853526600897, −10.61409086903879801098155257196, −9.924137585587717156972072207829, −8.015296704356235503051288184021, −7.23496285929140111960894151450, −6.12047555109775306282715572813, −5.10404555478529117292711421324, −3.55118943297317895987126285240, −1.61468379966502755398377391403,
0.24063037764214615929704032233, 1.78585102020162750830970000382, 3.54067393899106738242786373293, 5.01729577171976778559558554444, 6.45237114981399715862941951005, 7.03796784265886301339777818195, 8.432473790144297071846986422096, 9.998984603744714865147216000457, 11.05556892381785556672633739108, 11.53184459586165499115567254246