| L(s) = 1 | + (−1.21 − 4.84i)5-s + 11.5i·7-s + 15.2i·11-s + 17.0i·13-s + 10.4·17-s − 23.1·19-s − 13.9·23-s + (−22.0 + 11.8i)25-s − 44.8i·29-s + 5.44·31-s + (55.7 − 14.0i)35-s − 50.7i·37-s − 42.6i·41-s + 6.14i·43-s − 32.0·47-s + ⋯ |
| L(s) = 1 | + (−0.243 − 0.969i)5-s + 1.64i·7-s + 1.38i·11-s + 1.31i·13-s + 0.613·17-s − 1.21·19-s − 0.607·23-s + (−0.881 + 0.472i)25-s − 1.54i·29-s + 0.175·31-s + (1.59 − 0.400i)35-s − 1.37i·37-s − 1.04i·41-s + 0.142i·43-s − 0.682·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3157246056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3157246056\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.21 + 4.84i)T \) |
| good | 7 | \( 1 - 11.5iT - 49T^{2} \) |
| 11 | \( 1 - 15.2iT - 121T^{2} \) |
| 13 | \( 1 - 17.0iT - 169T^{2} \) |
| 17 | \( 1 - 10.4T + 289T^{2} \) |
| 19 | \( 1 + 23.1T + 361T^{2} \) |
| 23 | \( 1 + 13.9T + 529T^{2} \) |
| 29 | \( 1 + 44.8iT - 841T^{2} \) |
| 31 | \( 1 - 5.44T + 961T^{2} \) |
| 37 | \( 1 + 50.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.14iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 32.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 16.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 19.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 6.59T + 3.72e3T^{2} \) |
| 67 | \( 1 + 35.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 65.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 104. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 150.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 81.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 14.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 26.8iT - 9.40e3T^{2} \) |
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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
−9.430187772260904441034167696999, −8.915355455654324557406119342741, −8.221818623782906972991468710403, −7.33276819521069428735063683079, −6.25670335389427615604565149704, −5.56117030813241029656105140090, −4.60567743239365140950759482992, −4.02130634381681041126306871417, −2.30784182019987530311383108666, −1.82598980610013578801327305851,
0.086972295036330105246239051006, 1.20938512685302902777986279465, 2.99274793328313672789231782614, 3.46283995114875205046262949940, 4.40652436718268635572851964768, 5.64035967714144431619142697655, 6.47363259394888261971218711359, 7.15588043260133314962426646065, 8.033926964971310991863096178564, 8.437386369948228540358248319737
