| L(s) = 1 | + 1.41i·2-s − 1.73·3-s − 2.00·4-s + (4.59 + 1.96i)5-s − 2.44i·6-s + (6.81 − 1.57i)7-s − 2.82i·8-s + 2.99·9-s + (−2.77 + 6.50i)10-s − 8.15·11-s + 3.46·12-s + 14.6·13-s + (2.23 + 9.64i)14-s + (−7.96 − 3.40i)15-s + 4.00·16-s + 5.81·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.500·4-s + (0.919 + 0.393i)5-s − 0.408i·6-s + (0.974 − 0.225i)7-s − 0.353i·8-s + 0.333·9-s + (−0.277 + 0.650i)10-s − 0.741·11-s + 0.288·12-s + 1.12·13-s + (0.159 + 0.688i)14-s + (−0.530 − 0.226i)15-s + 0.250·16-s + 0.342·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.17741 + 0.986104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17741 + 0.986104i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (-4.59 - 1.96i)T \) |
| 7 | \( 1 + (-6.81 + 1.57i)T \) |
| good | 11 | \( 1 + 8.15T + 121T^{2} \) |
| 13 | \( 1 - 14.6T + 169T^{2} \) |
| 17 | \( 1 - 5.81T + 289T^{2} \) |
| 19 | \( 1 - 33.7iT - 361T^{2} \) |
| 23 | \( 1 - 37.2iT - 529T^{2} \) |
| 29 | \( 1 + 9.25T + 841T^{2} \) |
| 31 | \( 1 + 19.2iT - 961T^{2} \) |
| 37 | \( 1 + 63.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.25iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 23.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 71.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 42.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 34.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.99iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 38.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 124.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 56.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 90.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 16.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 82.7T + 9.40e3T^{2} \) |
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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
−12.47784456204500462600136337820, −11.17388565164930445188632084137, −10.46815122051645138434037471330, −9.447484361430528266533079603834, −8.120357222858832271093862120651, −7.28573696764433671691702131642, −5.81695855950818428806390750165, −5.51123960214425018001846410353, −3.85426210243658945714270614667, −1.59792226917437829863083949285,
1.10443463694290013294539137599, 2.57112856689765423782135269991, 4.60279990882651643198758192196, 5.32809588378804404888360188031, 6.55527919492423450444365334228, 8.244646608740651438584037633205, 9.017487275823878521432517788271, 10.28115542103915845350554153218, 10.92958617397967617679871605926, 11.79860513733746828799340106217
