| L(s) = 1 | + 3.63i·5-s + (1 − 2.44i)7-s + 1.50i·11-s − 5.91i·13-s − 5.14i·17-s − 5.24·19-s + 8.78i·23-s − 8.24·25-s − 8.91·29-s − 3.24·31-s + (8.91 + 3.63i)35-s + 37-s + 6.65i·41-s + 9.37i·43-s + 3.69·47-s + ⋯ |
| L(s) = 1 | + 1.62i·5-s + (0.377 − 0.925i)7-s + 0.454i·11-s − 1.64i·13-s − 1.24i·17-s − 1.20·19-s + 1.83i·23-s − 1.64·25-s − 1.65·29-s − 0.582·31-s + (1.50 + 0.615i)35-s + 0.164·37-s + 1.03i·41-s + 1.43i·43-s + 0.538·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2348839587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2348839587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
| good | 5 | \( 1 - 3.63iT - 5T^{2} \) |
| 11 | \( 1 - 1.50iT - 11T^{2} \) |
| 13 | \( 1 + 5.91iT - 13T^{2} \) |
| 17 | \( 1 + 5.14iT - 17T^{2} \) |
| 19 | \( 1 + 5.24T + 19T^{2} \) |
| 23 | \( 1 - 8.78iT - 23T^{2} \) |
| 29 | \( 1 + 8.91T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 6.65iT - 41T^{2} \) |
| 43 | \( 1 - 9.37iT - 43T^{2} \) |
| 47 | \( 1 - 3.69T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 8.91T + 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 10.8iT - 67T^{2} \) |
| 71 | \( 1 + 3.63iT - 71T^{2} \) |
| 73 | \( 1 + 0.420iT - 73T^{2} \) |
| 79 | \( 1 - 4.47iT - 79T^{2} \) |
| 83 | \( 1 + 3.69T + 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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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
−9.355967346641528020401138503511, −7.941300995750498387068471255125, −7.56478170158659667901970074088, −7.07182680301077184346309167881, −6.15680653620846991378733793454, −5.38017777466585760447485269800, −4.33551911647025977865853874290, −3.37937468640842531682727621720, −2.83243655251238924370354528644, −1.60579689076358203675037801330,
0.06780184692659005415246426677, 1.68130484232095230832576882433, 2.14174526118513057126558490459, 3.87942344716980959721899398316, 4.40879217110672421917994815551, 5.20704723750588350709998978738, 5.98127914031540064141535701313, 6.62981958143922836842261090879, 7.919511994803870624343131725835, 8.504777698765510255922699706942
