| L(s) = 1 | + (−1.36 − 0.361i)2-s + (−0.200 − 1.72i)3-s + (1.73 + 0.988i)4-s + (−2.41 + 1.69i)5-s + (−0.348 + 2.42i)6-s + (0.341 − 0.286i)7-s + (−2.02 − 1.97i)8-s + (−2.91 + 0.688i)9-s + (3.91 − 1.44i)10-s + (2.38 + 1.67i)11-s + (1.35 − 3.18i)12-s + (−0.0654 − 0.0305i)13-s + (−0.570 + 0.268i)14-s + (3.39 + 3.81i)15-s + (2.04 + 3.43i)16-s + (0.864 + 0.499i)17-s + ⋯ |
| L(s) = 1 | + (−0.966 − 0.255i)2-s + (−0.115 − 0.993i)3-s + (0.869 + 0.494i)4-s + (−1.08 + 0.756i)5-s + (−0.142 + 0.989i)6-s + (0.129 − 0.108i)7-s + (−0.714 − 0.699i)8-s + (−0.973 + 0.229i)9-s + (1.23 − 0.455i)10-s + (0.719 + 0.503i)11-s + (0.390 − 0.920i)12-s + (−0.0181 − 0.00846i)13-s + (−0.152 + 0.0717i)14-s + (0.876 + 0.986i)15-s + (0.511 + 0.859i)16-s + (0.209 + 0.121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.678514 + 0.0385938i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678514 + 0.0385938i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.361i)T \) |
| 3 | \( 1 + (0.200 + 1.72i)T \) |
| good | 5 | \( 1 + (2.41 - 1.69i)T + (1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (-0.341 + 0.286i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-2.38 - 1.67i)T + (3.76 + 10.3i)T^{2} \) |
| 13 | \( 1 + (0.0654 + 0.0305i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.864 - 0.499i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.86 - 1.84i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (4.36 - 5.20i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-8.22 + 3.83i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-0.0632 + 0.0753i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.10 - 4.10i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.96 + 2.16i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-5.85 + 8.36i)T + (-14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (3.58 - 3.01i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.597 - 0.597i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.19 - 10.2i)T + (-20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (4.90 + 0.429i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (-1.12 + 2.41i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-3.05 - 1.76i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.79 - 1.61i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.54 - 12.4i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.58 + 7.68i)T + (-53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (1.77 + 3.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.441 - 2.50i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.39414975726365542013533340473, −10.33244685868446684002784224688, −9.346969458927036363057414369533, −8.117010025570739748218075360806, −7.57599946177540503456561154049, −6.95516256961420402568234512334, −5.89991114233416511685147961129, −3.86885740484219512607977623252, −2.75606838020628514205076404441, −1.20737960157576839471502518236,
0.72620775167046206134968468873, 3.05948415981615463815914543222, 4.33853058934431817698764758252, 5.36798953500217263802242712549, 6.52798372915565878857857642558, 7.84549735999856636533851938760, 8.506944804995343755764298381234, 9.239833766664855521785797975383, 10.06112811441090623662758369079, 11.13054109678675435234410336323
