| L(s) = 1 | + (1.93 + 1.11i)5-s + (3.57 + 6.18i)7-s + (−2.20 + 1.27i)11-s + (−5.57 + 9.66i)13-s + 2.05i·17-s − 19.3·19-s + (−10.4 − 6.01i)23-s + (2.5 + 4.33i)25-s + (−19.3 + 11.1i)29-s + (−7.08 + 12.2i)31-s + 15.9i·35-s − 46.3·37-s + (15.0 + 8.70i)41-s + (−13.5 − 23.3i)43-s + (42.2 − 24.3i)47-s + ⋯ |
| L(s) = 1 | + (0.387 + 0.223i)5-s + (0.510 + 0.884i)7-s + (−0.200 + 0.115i)11-s + (−0.429 + 0.743i)13-s + 0.121i·17-s − 1.01·19-s + (−0.453 − 0.261i)23-s + (0.100 + 0.173i)25-s + (−0.665 + 0.384i)29-s + (−0.228 + 0.395i)31-s + 0.456i·35-s − 1.25·37-s + (0.367 + 0.212i)41-s + (−0.314 − 0.544i)43-s + (0.898 − 0.519i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7269169032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7269169032\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.93 - 1.11i)T \) |
| good | 7 | \( 1 + (-3.57 - 6.18i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (2.20 - 1.27i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (5.57 - 9.66i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 2.05iT - 289T^{2} \) |
| 19 | \( 1 + 19.3T + 361T^{2} \) |
| 23 | \( 1 + (10.4 + 6.01i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (19.3 - 11.1i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (7.08 - 12.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 46.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-15.0 - 8.70i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (13.5 + 23.3i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-42.2 + 24.3i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 9.84iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (76.1 + 43.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (29.9 + 51.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-3.85 + 6.68i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 79.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 6.01T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-8.46 - 14.6i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-127. + 73.4i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 100. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (36.3 + 62.8i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
| 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
−9.455026034841123933263980625821, −8.848947829971338031639125002052, −8.126299148722859952246518943258, −7.12228094171189521348378852322, −6.37286614704024017001610982622, −5.48000383873849278065571877790, −4.76128459296708504096715171398, −3.66918224098210086960058458915, −2.37267320673731032636108237991, −1.78181874066654047672686160884,
0.18029493307839095962707118944, 1.46818806092107756120806199081, 2.58073393229865040142740711618, 3.82178412339511229264686326792, 4.63447096126449324824040491595, 5.52266516335942224261654801656, 6.33441860634068595320440893142, 7.43690911785702022794016509491, 7.87309489118915182619953051343, 8.844612528849019816978947961073
