L(s) = 1 | + (0.987 − 0.156i)2-s + (−1.49 − 0.876i)3-s + (0.951 − 0.309i)4-s + (0.197 − 2.22i)5-s + (−1.61 − 0.632i)6-s + (−1.43 − 1.43i)7-s + (0.891 − 0.453i)8-s + (1.46 + 2.61i)9-s + (−0.152 − 2.23i)10-s + (3.22 − 4.44i)11-s + (−1.69 − 0.372i)12-s + (−1.00 + 6.34i)13-s + (−1.63 − 1.19i)14-s + (−2.24 + 3.15i)15-s + (0.809 − 0.587i)16-s + (0.101 + 0.199i)17-s + ⋯ |
L(s) = 1 | + (0.698 − 0.110i)2-s + (−0.862 − 0.506i)3-s + (0.475 − 0.154i)4-s + (0.0885 − 0.996i)5-s + (−0.658 − 0.258i)6-s + (−0.541 − 0.541i)7-s + (0.315 − 0.160i)8-s + (0.487 + 0.873i)9-s + (−0.0483 − 0.705i)10-s + (0.973 − 1.33i)11-s + (−0.488 − 0.107i)12-s + (−0.278 + 1.75i)13-s + (−0.437 − 0.318i)14-s + (−0.580 + 0.814i)15-s + (0.202 − 0.146i)16-s + (0.0246 + 0.0483i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00962 - 0.771526i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00962 - 0.771526i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.987 + 0.156i)T \) |
| 3 | \( 1 + (1.49 + 0.876i)T \) |
| 5 | \( 1 + (-0.197 + 2.22i)T \) |
good | 7 | \( 1 + (1.43 + 1.43i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.22 + 4.44i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.00 - 6.34i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.101 - 0.199i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.05 - 0.669i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.339 - 2.14i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-2.33 - 7.19i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.944 - 2.90i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.46 - 1.02i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (0.896 + 1.23i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.14 - 2.14i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.66 + 2.37i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-5.56 + 10.9i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (4.61 - 3.35i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.55 - 1.85i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.22 + 1.13i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-5.77 + 1.87i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.02 - 0.478i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (11.0 - 3.57i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (9.31 - 4.74i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (7.34 + 5.33i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.09 + 2.14i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−12.81635651213844940869564097992, −11.79566730068451608291828940163, −11.33510653048209132320017558340, −9.867794723916756012784981484824, −8.678322406311410774001343232315, −7.05114784918168813412579090324, −6.23840429676993303930912946781, −5.05599083348693592222715048983, −3.84751562341559079706067230388, −1.35012968792871053763405152114,
2.86536517155134378996416611918, 4.28121878371232245021454613339, 5.66463449373361312195821417183, 6.48337219455467138049395355467, 7.54898858636538741800557217879, 9.605799706230534896209238336695, 10.23061001275014000420090255019, 11.38135798085094176521861407802, 12.21962982106789682610306109827, 12.99296395551321175837651705344