| L(s) = 1 | + (1.19 − 0.866i)2-s + (−2.65 + 1.18i)3-s + (0.0532 − 0.163i)4-s + (0.5 + 0.866i)5-s + (−2.14 + 3.70i)6-s + (2.31 + 2.57i)7-s + (0.832 + 2.56i)8-s + (3.64 − 4.04i)9-s + (1.34 + 0.599i)10-s + (−2.60 + 0.554i)11-s + (0.0523 + 0.498i)12-s + (0.147 − 1.40i)13-s + (4.98 + 1.05i)14-s + (−2.35 − 1.70i)15-s + (3.49 + 2.53i)16-s + (−2.60 − 0.554i)17-s + ⋯ |
| L(s) = 1 | + (0.843 − 0.612i)2-s + (−1.53 + 0.682i)3-s + (0.0266 − 0.0819i)4-s + (0.223 + 0.387i)5-s + (−0.874 + 1.51i)6-s + (0.874 + 0.971i)7-s + (0.294 + 0.905i)8-s + (1.21 − 1.34i)9-s + (0.425 + 0.189i)10-s + (−0.786 + 0.167i)11-s + (0.0151 + 0.143i)12-s + (0.0408 − 0.388i)13-s + (1.33 + 0.283i)14-s + (−0.606 − 0.441i)15-s + (0.872 + 0.634i)16-s + (−0.632 − 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01058 + 0.491407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01058 + 0.491407i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-1.63 + 5.32i)T \) |
| good | 2 | \( 1 + (-1.19 + 0.866i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (2.65 - 1.18i)T + (2.00 - 2.22i)T^{2} \) |
| 7 | \( 1 + (-2.31 - 2.57i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (2.60 - 0.554i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.147 + 1.40i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (2.60 + 0.554i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.482 + 4.59i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-2.45 - 7.56i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-7.61 + 5.53i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-4.77 + 8.26i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.186 + 0.0830i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.336 - 3.20i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-4.52 - 3.29i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.59 - 2.88i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (9.59 - 4.27i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 + (2.34 + 4.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.37 + 7.08i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-1.49 + 0.318i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-9.67 - 2.05i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (6.38 + 2.84i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (4.05 - 12.4i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.59 + 14.1i)T + (-78.4 - 57.0i)T^{2} \) |
| 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
−12.83267033588779625455639356970, −11.86791226309812707602007583913, −11.25360239255983368436326199780, −10.69688283273608465081810639218, −9.336291932734072365321341054391, −7.77662525972786268763141903422, −6.02306069050257704323424845250, −5.19651646780220855486669668989, −4.44310227918571940835390811737, −2.58308710302645168495416677862,
1.14899028389776651341282266271, 4.53874138246059304533032490924, 5.06763969461199225964316375655, 6.28537200182313949098812847491, 6.97346889528446387165876270838, 8.228847117960523175594317509366, 10.34414005048110043344296150139, 10.78751007429654788651745713413, 12.11957414393279908133605718032, 12.84519807984946123132021667479
