L(s) = 1 | + (0.153 − 0.192i)2-s + (−1.67 − 0.804i)3-s + (0.431 + 1.89i)4-s + (0.217 + 0.104i)5-s + (−0.410 + 0.197i)6-s + (−2.59 − 0.499i)7-s + (0.873 + 0.420i)8-s + (0.271 + 0.340i)9-s + (0.0536 − 0.0258i)10-s + (−0.00113 + 0.00142i)11-s + (0.799 − 3.50i)12-s + (0.623 − 0.781i)13-s + (−0.494 + 0.423i)14-s + (−0.279 − 0.350i)15-s + (−3.27 + 1.57i)16-s + (1.38 − 6.06i)17-s + ⋯ |
L(s) = 1 | + (0.108 − 0.136i)2-s + (−0.964 − 0.464i)3-s + (0.215 + 0.945i)4-s + (0.0974 + 0.0469i)5-s + (−0.167 + 0.0807i)6-s + (−0.981 − 0.188i)7-s + (0.308 + 0.148i)8-s + (0.0905 + 0.113i)9-s + (0.0169 − 0.00816i)10-s + (−0.000342 + 0.000429i)11-s + (0.230 − 1.01i)12-s + (0.172 − 0.216i)13-s + (−0.132 + 0.113i)14-s + (−0.0721 − 0.0904i)15-s + (−0.819 + 0.394i)16-s + (0.335 − 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.668847 - 0.566732i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.668847 - 0.566732i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.59 + 0.499i)T \) |
| 13 | \( 1 + (-0.623 + 0.781i)T \) |
good | 2 | \( 1 + (-0.153 + 0.192i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (1.67 + 0.804i)T + (1.87 + 2.34i)T^{2} \) |
| 5 | \( 1 + (-0.217 - 0.104i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (0.00113 - 0.00142i)T + (-2.44 - 10.7i)T^{2} \) |
| 17 | \( 1 + (-1.38 + 6.06i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 + (1.18 + 5.20i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (0.125 - 0.550i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 - 9.65T + 31T^{2} \) |
| 37 | \( 1 + (-1.63 + 7.16i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (9.56 + 4.60i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (3.15 - 1.52i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-5.99 + 7.51i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (0.930 + 4.07i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (2.09 - 1.00i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-0.251 + 1.10i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + 5.22T + 67T^{2} \) |
| 71 | \( 1 + (-0.583 - 2.55i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.06 + 5.10i)T + (-16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 - 0.670T + 79T^{2} \) |
| 83 | \( 1 + (-7.51 - 9.42i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (9.78 + 12.2i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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
−10.47683390964656515542621586036, −9.656736075502128726651599433795, −8.588527379544852194440776575343, −7.44143528145403242407521868382, −6.85676960157564038235480317675, −6.00287376679626588850562355861, −4.92958312817787969039010385651, −3.55695210286730476572641695430, −2.66433509083248418962364609805, −0.56010057257215692930628145526,
1.36256692260714262802518441333, 3.18139684256286275337368533596, 4.52993211345875248607639432231, 5.62385198058605073882077868202, 5.94902664969694814934186300989, 6.87155335216403172423919455168, 8.113973407402111580454029347622, 9.556520821283198102653413449311, 9.900531845833704683107022079296, 10.66777027438668112110522761237