| L(s) = 1 | + (−0.987 − 0.156i)2-s + (−1.36 − 1.06i)3-s + (0.951 + 0.309i)4-s + (−1.48 + 1.66i)5-s + (1.18 + 1.26i)6-s + (−1.08 + 1.08i)7-s + (−0.891 − 0.453i)8-s + (0.738 + 2.90i)9-s + (1.73 − 1.41i)10-s + (1.61 + 2.21i)11-s + (−0.971 − 1.43i)12-s + (0.355 + 2.24i)13-s + (1.24 − 0.903i)14-s + (3.80 − 0.697i)15-s + (0.809 + 0.587i)16-s + (−2.77 + 5.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 − 0.110i)2-s + (−0.789 − 0.613i)3-s + (0.475 + 0.154i)4-s + (−0.665 + 0.746i)5-s + (0.483 + 0.516i)6-s + (−0.410 + 0.410i)7-s + (−0.315 − 0.160i)8-s + (0.246 + 0.969i)9-s + (0.547 − 0.447i)10-s + (0.486 + 0.669i)11-s + (−0.280 − 0.413i)12-s + (0.0985 + 0.622i)13-s + (0.332 − 0.241i)14-s + (0.983 − 0.180i)15-s + (0.202 + 0.146i)16-s + (−0.673 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0494 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0494 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.267825 + 0.281415i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267825 + 0.281415i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.36 + 1.06i)T \) |
| 5 | \( 1 + (1.48 - 1.66i)T \) |
| good | 7 | \( 1 + (1.08 - 1.08i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.61 - 2.21i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.355 - 2.24i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.77 - 5.44i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (4.05 - 1.31i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.805 + 5.08i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (2.37 - 7.29i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.85 + 5.71i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.29 - 0.521i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-5.36 + 7.38i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-6.65 - 6.65i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.15 - 1.09i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (0.199 + 0.391i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (5.86 + 4.26i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.14 - 5.91i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-5.28 - 2.69i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-5.89 - 1.91i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-10.2 - 1.62i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-6.06 - 1.97i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.92 - 3.52i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-8.12 + 5.90i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.51 - 10.8i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.71763541669145016721005378025, −12.29197394666675365078513898428, −11.05318857914161001041506764511, −10.61262563568872140459570978045, −9.134546188827949681951519043822, −7.943614028370373115443613078336, −6.81650019101581716833199436092, −6.21521594154342623230882625796, −4.16345342651908062055323637181, −2.12991623717074618903063094974,
0.51502484956680451990460462532, 3.63633197346087888200747905167, 4.99776534474108153405960869742, 6.30497862824900151202547495454, 7.50546290238613027769154099985, 8.859628461155013172145527093196, 9.568605257251211845589628929837, 10.84775456409705574807027272131, 11.47319329175925525507138429308, 12.47738419692781084486705490102
