| L(s) = 1 | + (−0.998 − 1.19i)2-s + (0.173 + 0.984i)3-s + (−0.0717 + 0.407i)4-s + (−4.35 + 0.767i)5-s + (0.998 − 1.19i)6-s + (2.61 − 0.431i)7-s + (−2.13 + 1.23i)8-s + (−0.939 + 0.342i)9-s + (5.26 + 4.41i)10-s + 4.43·11-s − 0.413·12-s + (1.79 + 1.50i)13-s + (−3.12 − 2.67i)14-s + (−1.51 − 4.15i)15-s + (4.37 + 1.59i)16-s + (2.32 − 6.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.706 − 0.841i)2-s + (0.100 + 0.568i)3-s + (−0.0358 + 0.203i)4-s + (−1.94 + 0.343i)5-s + (0.407 − 0.485i)6-s + (0.986 − 0.163i)7-s + (−0.754 + 0.435i)8-s + (−0.313 + 0.114i)9-s + (1.66 + 1.39i)10-s + 1.33·11-s − 0.119·12-s + (0.498 + 0.417i)13-s + (−0.833 − 0.715i)14-s + (−0.390 − 1.07i)15-s + (1.09 + 0.398i)16-s + (0.564 − 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.732889 - 0.312968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.732889 - 0.312968i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 7 | \( 1 + (-2.61 + 0.431i)T \) |
| 19 | \( 1 + (3.60 + 2.45i)T \) |
| good | 2 | \( 1 + (0.998 + 1.19i)T + (-0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (4.35 - 0.767i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 + (-1.79 - 1.50i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.32 + 6.39i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-3.21 - 2.69i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-6.51 - 1.14i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.156 - 0.270i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.947 - 0.547i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.77 + 2.32i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.819 - 0.298i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.76 - 4.84i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.778 - 0.137i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (2.36 + 0.861i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.0104 - 0.0124i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (4.31 - 5.13i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.82 + 10.5i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (3.96 - 0.698i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (0.699 - 1.92i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.10 - 3.52i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.505 - 2.86i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.03 - 5.85i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.18135332117369065516705752714, −10.58402288741734179833947963612, −9.214389103906138191385635874693, −8.695001169699288174879855798710, −7.73096706348584956084870792463, −6.70884659286413186067055026806, −4.87970912092169299954973743054, −3.99315837250438163156883893720, −2.93575723213642528442225981047, −0.948650262470372003725792988084,
1.05516363696183186086615411693, 3.49599738030402495793693068961, 4.36519512283155404206033454211, 6.06865752175898557104229780972, 7.01445578283873815910467930135, 7.911784047172151986824376546867, 8.426996678664922782822373936832, 8.795840959743516571027609273221, 10.60285640487666753492526052567, 11.56856456116279240830668845192
