L(s) = 1 | + (−0.148 − 0.257i)2-s + (0.310 − 1.70i)3-s + (0.955 − 1.65i)4-s − 5-s + (−0.485 + 0.173i)6-s + (−2.64 + 0.0857i)7-s − 1.16·8-s + (−2.80 − 1.05i)9-s + (0.148 + 0.257i)10-s + 4.84·11-s + (−2.52 − 2.14i)12-s + (−2.59 − 4.49i)13-s + (0.415 + 0.668i)14-s + (−0.310 + 1.70i)15-s + (−1.73 − 3.01i)16-s + (1.80 + 3.11i)17-s + ⋯ |
L(s) = 1 | + (−0.105 − 0.182i)2-s + (0.179 − 0.983i)3-s + (0.477 − 0.827i)4-s − 0.447·5-s + (−0.198 + 0.0707i)6-s + (−0.999 + 0.0324i)7-s − 0.411·8-s + (−0.935 − 0.353i)9-s + (0.0470 + 0.0814i)10-s + 1.46·11-s + (−0.728 − 0.618i)12-s + (−0.719 − 1.24i)13-s + (0.111 + 0.178i)14-s + (−0.0802 + 0.439i)15-s + (−0.434 − 0.752i)16-s + (0.436 + 0.756i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.327493 - 1.04456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.327493 - 1.04456i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.310 + 1.70i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2.64 - 0.0857i)T \) |
good | 2 | \( 1 + (0.148 + 0.257i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 - 4.84T + 11T^{2} \) |
| 13 | \( 1 + (2.59 + 4.49i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.80 - 3.11i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.03 - 3.51i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.983T + 23T^{2} \) |
| 29 | \( 1 + (-2.30 + 3.98i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.60 + 6.23i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.25 + 5.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.298 + 0.516i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.0565 + 0.0979i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.57 + 6.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.78 - 8.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.75 + 6.49i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.49 - 12.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.78 + 6.55i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + (1.67 + 2.90i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.59 + 4.49i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.45 - 9.45i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.64 - 13.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.85 + 11.8i)T + (-48.5 - 84.0i)T^{2} \) |
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
−11.48256632371641572258355216612, −10.29335486208915066197638454790, −9.534331517667855329623470246018, −8.375795183044322007205343546722, −7.30419158765127314124773820985, −6.36562116234720203143451448284, −5.74997578482884952165307112929, −3.72922188185437089919203994218, −2.42447528801500510678806094678, −0.802136450532578940710193037975,
2.79016273355352793584093706263, 3.73730456748897050130633203833, 4.71059909770432977902921753989, 6.53891235071652500996475173641, 7.02108749585440974368298738827, 8.508646847140913161443479345979, 9.181159501033769489045573908417, 9.956617334405983484116081746138, 11.37131006819159391097603270962, 11.77567179925229645485246789163