L(s) = 1 | − 2.82i·3-s + (4.89 − i)5-s − 8.48·7-s + 0.999·9-s − 13.8·11-s − 9.79·13-s + (−2.82 − 13.8i)15-s + 19.5i·17-s − 13.8·19-s + 24i·21-s + 25.4·23-s + (22.9 − 9.79i)25-s − 28.2i·27-s + 22i·29-s + 55.4i·31-s + ⋯ |
L(s) = 1 | − 0.942i·3-s + (0.979 − 0.200i)5-s − 1.21·7-s + 0.111·9-s − 1.25·11-s − 0.753·13-s + (−0.188 − 0.923i)15-s + 1.15i·17-s − 0.729·19-s + 1.14i·21-s + 1.10·23-s + (0.919 − 0.391i)25-s − 1.04i·27-s + 0.758i·29-s + 1.78i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.071441664\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.071441664\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-4.89 + i)T \) |
good | 3 | \( 1 + 2.82iT - 9T^{2} \) |
| 7 | \( 1 + 8.48T + 49T^{2} \) |
| 11 | \( 1 + 13.8T + 121T^{2} \) |
| 13 | \( 1 + 9.79T + 169T^{2} \) |
| 17 | \( 1 - 19.5iT - 289T^{2} \) |
| 19 | \( 1 + 13.8T + 361T^{2} \) |
| 23 | \( 1 - 25.4T + 529T^{2} \) |
| 29 | \( 1 - 22iT - 841T^{2} \) |
| 31 | \( 1 - 55.4iT - 961T^{2} \) |
| 37 | \( 1 - 48.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 22T + 1.68e3T^{2} \) |
| 43 | \( 1 - 59.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 8.48T + 2.20e3T^{2} \) |
| 53 | \( 1 - 29.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 13.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 46iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 59.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 27.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 78.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 76.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 146T + 7.92e3T^{2} \) |
| 97 | \( 1 - 58.7iT - 9.40e3T^{2} \) |
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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
−9.724250032066039220390504858090, −8.770290379649794972352531372286, −7.935457121414520631481477870886, −6.90456998037570117876206809297, −6.50048946812070170746792163834, −5.57940359300623156262651417608, −4.66228256784138372501145439982, −3.10981992344887429938853796903, −2.32044046092389322388733498034, −1.18638932925301745769951804802,
0.31408021535444123282075995387, 2.39135980402347398176459310606, 2.97767492948325278309489420319, 4.25753534942235617177706271559, 5.13525608068009937188091866808, 5.86356963686802688401791055477, 6.86125197757352229260580040720, 7.58010350845751829869285012654, 8.959563458054190926653937475224, 9.569593456041841004682602590343