L(s) = 1 | + 2.51i·2-s + (−0.170 − 1.72i)3-s − 4.34·4-s + (−0.5 + 0.866i)5-s + (4.34 − 0.430i)6-s + (1.80 + 1.93i)7-s − 5.90i·8-s + (−2.94 + 0.588i)9-s + (−2.18 − 1.25i)10-s + (−5.56 + 3.21i)11-s + (0.742 + 7.48i)12-s + (−1.77 + 1.02i)13-s + (−4.86 + 4.55i)14-s + (1.57 + 0.713i)15-s + 6.18·16-s + (−1.68 + 2.92i)17-s + ⋯ |
L(s) = 1 | + 1.78i·2-s + (−0.0986 − 0.995i)3-s − 2.17·4-s + (−0.223 + 0.387i)5-s + (1.77 − 0.175i)6-s + (0.682 + 0.730i)7-s − 2.08i·8-s + (−0.980 + 0.196i)9-s + (−0.689 − 0.398i)10-s + (−1.67 + 0.969i)11-s + (0.214 + 2.16i)12-s + (−0.491 + 0.283i)13-s + (−1.30 + 1.21i)14-s + (0.407 + 0.184i)15-s + 1.54·16-s + (−0.409 + 0.709i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.128115 - 0.669921i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.128115 - 0.669921i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.170 + 1.72i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-1.80 - 1.93i)T \) |
good | 2 | \( 1 - 2.51iT - 2T^{2} \) |
| 11 | \( 1 + (5.56 - 3.21i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.77 - 1.02i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.68 - 2.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.51 + 2.02i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.48 + 1.43i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.49 - 3.17i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.965iT - 31T^{2} \) |
| 37 | \( 1 + (1.31 + 2.27i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.50 - 6.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.788 + 1.36i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6.05T + 47T^{2} \) |
| 53 | \( 1 + (3.22 + 1.86i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 7.33iT - 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 5.13iT - 71T^{2} \) |
| 73 | \( 1 + (9.61 + 5.55i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 3.85T + 79T^{2} \) |
| 83 | \( 1 + (-0.549 + 0.951i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.423 + 0.733i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.96 + 1.71i)T + (48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.54147602504398768460348049128, −11.42975595511912758405087537922, −10.09565350289232375825299730933, −8.740459110041245248959055661513, −8.004666831245109070200048067206, −7.39838859129958067578067403011, −6.56055216970086454455965697302, −5.45741087091914804440195110616, −4.76859934751154887421737269743, −2.43697034087067646846117490490,
0.48789230235841854425808979930, 2.60176859444445946475042062562, 3.64926904681790041838061204384, 4.74955661897024946328890193343, 5.38090755631148007335796954106, 7.86349629336288073633413608549, 8.591588584273798615299082561042, 9.843148363577533472760498126635, 10.27273100522643904375101621485, 11.19428515596925884212790327080