| L(s) = 1 | + (−0.557 + 1.53i)2-s + (−0.844 − 1.51i)3-s + (−0.500 − 0.420i)4-s + (−1.06 + 1.96i)5-s + (2.78 − 0.451i)6-s + (1.77 + 2.11i)7-s + (−1.89 + 1.09i)8-s + (−1.57 + 2.55i)9-s + (−2.41 − 2.72i)10-s + (0.0600 + 0.340i)11-s + (−0.212 + 1.11i)12-s + (1.40 + 3.85i)13-s + (−4.23 + 1.54i)14-s + (3.87 − 0.0536i)15-s + (−0.847 − 4.80i)16-s + (0.713 + 0.411i)17-s + ⋯ |
| L(s) = 1 | + (−0.393 + 1.08i)2-s + (−0.487 − 0.872i)3-s + (−0.250 − 0.210i)4-s + (−0.475 + 0.879i)5-s + (1.13 − 0.184i)6-s + (0.672 + 0.800i)7-s + (−0.671 + 0.387i)8-s + (−0.524 + 0.851i)9-s + (−0.764 − 0.861i)10-s + (0.0181 + 0.102i)11-s + (−0.0612 + 0.321i)12-s + (0.388 + 1.06i)13-s + (−1.13 + 0.411i)14-s + (0.999 − 0.0138i)15-s + (−0.211 − 1.20i)16-s + (0.172 + 0.0998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.348065 + 0.634729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.348065 + 0.634729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.844 + 1.51i)T \) |
| 5 | \( 1 + (1.06 - 1.96i)T \) |
| good | 2 | \( 1 + (0.557 - 1.53i)T + (-1.53 - 1.28i)T^{2} \) |
| 7 | \( 1 + (-1.77 - 2.11i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.0600 - 0.340i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.40 - 3.85i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.713 - 0.411i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.61 + 4.52i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.95 + 3.51i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-6.53 - 2.37i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.66 - 4.75i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (4.91 + 2.84i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.88 - 1.05i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.26 + 0.400i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.73 - 4.45i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 12.6iT - 53T^{2} \) |
| 59 | \( 1 + (-1.74 + 9.92i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (9.63 - 8.08i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.08 - 8.48i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.949 + 1.64i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.359 - 0.207i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.5 - 4.93i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.03 - 5.58i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-8.98 - 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.81 + 0.673i)T + (91.1 - 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.92002037480265444292117895311, −12.31820584311684993598027169088, −11.63866624052694994588458698751, −10.77841567434312319480509717952, −8.805263989230204355984716601193, −8.136632562211743978844001795527, −6.85240454643918081761089654513, −6.51729292998415624188362363643, −5.02393765561559346513258137833, −2.52657471047131696994975280262,
0.955947148423783056691865725614, 3.46650278851868322961626840467, 4.59467318831681543187122644934, 5.96370165008467461009334181365, 7.945165718336178389861849057167, 8.980076581625946130099945399103, 10.18482007506325101073373706650, 10.71699824830141934144806932291, 11.72068126553414636925769850812, 12.37601466297324421905626711875
