L(s) = 1 | + (1.17 + 0.788i)2-s + (0.758 + 1.85i)4-s + (−0.386 − 2.20i)5-s + (1.51 − 1.51i)7-s + (−0.568 + 2.77i)8-s + (1.28 − 2.89i)10-s + 3.92·11-s + (3.56 + 3.56i)13-s + (2.97 − 0.585i)14-s + (−2.85 + 2.80i)16-s + (−1.37 − 1.37i)17-s + 4i·19-s + (3.78 − 2.38i)20-s + (4.60 + 3.09i)22-s + (−5.17 − 5.17i)23-s + ⋯ |
L(s) = 1 | + (0.830 + 0.557i)2-s + (0.379 + 0.925i)4-s + (−0.172 − 0.984i)5-s + (0.573 − 0.573i)7-s + (−0.200 + 0.979i)8-s + (0.405 − 0.914i)10-s + 1.18·11-s + (0.988 + 0.988i)13-s + (0.795 − 0.156i)14-s + (−0.712 + 0.701i)16-s + (−0.333 − 0.333i)17-s + 0.917i·19-s + (0.846 − 0.533i)20-s + (0.982 + 0.659i)22-s + (−1.07 − 1.07i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.15604 + 0.579211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.15604 + 0.579211i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.17 - 0.788i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.386 + 2.20i)T \) |
good | 7 | \( 1 + (-1.51 + 1.51i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 + (-3.56 - 3.56i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.37 + 1.37i)T + 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (5.17 + 5.17i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.95T + 29T^{2} \) |
| 31 | \( 1 + 7.12iT - 31T^{2} \) |
| 37 | \( 1 + (3.56 - 3.56i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.75T + 41T^{2} \) |
| 43 | \( 1 + (-5.40 + 5.40i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.54 - 1.54i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.81 + 1.81i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.92iT - 59T^{2} \) |
| 61 | \( 1 - 13.1iT - 61T^{2} \) |
| 67 | \( 1 + (5.40 + 5.40i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 + (6.67 - 6.67i)T - 83iT^{2} \) |
| 89 | \( 1 - 18.4iT - 89T^{2} \) |
| 97 | \( 1 + (10.4 + 10.4i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.75687372308894300012410063117, −10.98693606632160998576872215538, −9.388338948486197499733229742916, −8.545040805429322670168180992564, −7.72616564304433268375289904846, −6.57489690001677347825723768443, −5.68366817251961730240019879157, −4.18284713205037151134031267811, −4.08192159048372531139262465458, −1.72098773052178247465044446417,
1.75249608383752038717588724173, 3.17864467275976402677148013726, 4.06633470121126682078310667973, 5.52971677656390901840339748716, 6.29916664799389268772735248662, 7.35893353073280717285010119125, 8.697113758084946467308186206686, 9.764766965555861392439342056860, 10.89637687240198607005605828636, 11.29024024587278309337189533984