| 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
−12.99296395551321175837651705344, −12.21962982106789682610306109827, −11.38135798085094176521861407802, −10.23061001275014000420090255019, −9.605799706230534896209238336695, −7.54898858636538741800557217879, −6.48337219455467138049395355467, −5.66463449373361312195821417183, −4.28121878371232245021454613339, −2.86536517155134378996416611918,
1.35012968792871053763405152114, 3.84751562341559079706067230388, 5.05599083348693592222715048983, 6.23840429676993303930912946781, 7.05114784918168813412579090324, 8.678322406311410774001343232315, 9.867794723916756012784981484824, 11.33510653048209132320017558340, 11.79566730068451608291828940163, 12.81635651213844940869564097992
