| L(s) = 1 | − 3.44i·3-s − 8.89·9-s − 5.44i·11-s + 1.89·17-s − 6.34i·19-s + 20.3i·27-s − 18.7·33-s − 6.79·41-s + 10i·43-s − 7·49-s − 6.55i·51-s − 21.8·57-s + 6i·59-s − 0.348i·67-s + 15.6·73-s + ⋯ |
| L(s) = 1 | − 1.99i·3-s − 2.96·9-s − 1.64i·11-s + 0.460·17-s − 1.45i·19-s + 3.91i·27-s − 3.27·33-s − 1.06·41-s + 1.52i·43-s − 49-s − 0.917i·51-s − 2.90·57-s + 0.781i·59-s − 0.0425i·67-s + 1.83·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.026389232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026389232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.44iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 5.44iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 + 6.34iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 0.348iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6.55iT - 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.540339703953776210074834015992, −8.137744723266701735495628161361, −7.26294383048984326556110567060, −6.53455498020109620982278037090, −5.93995314542723740075259388663, −5.08579881745749294888630438122, −3.30173201117611897198000901397, −2.65819863299756873929878575094, −1.37891358579616000913528935289, −0.40141296731276892723743300050,
2.11322504685344925314654835613, 3.40729810453939568398121871495, 4.02655996405017393432138166371, 4.90898720589016166637762190945, 5.45699583734831101347793920249, 6.51455866753393542790214056899, 7.75009970375648314632341689773, 8.512194536439929949140408726198, 9.403983237483167233877710890904, 9.996769617314878661395607678662
