| L(s) = 1 | + (−1.45 + 0.946i)3-s − 5-s + 1.74i·7-s + (1.20 − 2.74i)9-s + 0.702·11-s + 4.70i·13-s + (1.45 − 0.946i)15-s − 1.49i·17-s − 1.07·19-s + (−1.64 − 2.52i)21-s − 2.17i·23-s + 25-s + (0.849 + 5.12i)27-s + 2.17i·29-s + 3.66i·31-s + ⋯ |
| L(s) = 1 | + (−0.837 + 0.546i)3-s − 0.447·5-s + 0.657i·7-s + (0.402 − 0.915i)9-s + 0.211·11-s + 1.30i·13-s + (0.374 − 0.244i)15-s − 0.362i·17-s − 0.247·19-s + (−0.359 − 0.550i)21-s − 0.452i·23-s + 0.200·25-s + (0.163 + 0.986i)27-s + 0.404i·29-s + 0.658i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4340116469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4340116469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.45 - 0.946i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-7.89 - 2.15i)T \) |
| good | 7 | \( 1 - 1.74iT - 7T^{2} \) |
| 11 | \( 1 - 0.702T + 11T^{2} \) |
| 13 | \( 1 - 4.70iT - 13T^{2} \) |
| 17 | \( 1 + 1.49iT - 17T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 23 | \( 1 + 2.17iT - 23T^{2} \) |
| 29 | \( 1 - 2.17iT - 29T^{2} \) |
| 31 | \( 1 - 3.66iT - 31T^{2} \) |
| 37 | \( 1 + 7.97T + 37T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 + 2.43iT - 43T^{2} \) |
| 47 | \( 1 - 8.94iT - 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 9.27iT - 59T^{2} \) |
| 61 | \( 1 - 2.37iT - 61T^{2} \) |
| 71 | \( 1 + 15.2iT - 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 - 2.10iT - 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 9.88iT - 89T^{2} \) |
| 97 | \( 1 - 10.0iT - 97T^{2} \) |
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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
−9.046567436505444364582007995305, −8.272276379441344101552142642702, −7.02505533115567604533848039353, −6.76572939370322277454003323447, −5.79638335301623375485588654240, −5.11231337256878616053956152644, −4.34321108554454198256029625576, −3.70562596940197651372284564916, −2.56636467006887971219080263636, −1.32716601695427627895179727704,
0.16536288990080992961990278878, 1.10327512048245843221287737154, 2.28021602768356307921172921211, 3.51958276354728954720997720799, 4.21523938262721593120636970852, 5.24950488868449123922161519540, 5.72406363591422224541690110705, 6.73886300788629605169705090605, 7.18633746418875674432008870090, 8.009524803587890863082077708259
