L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.280 + 1.59i)5-s + (0.500 − 0.866i)8-s + (0.766 − 0.642i)9-s + (1.23 − 1.04i)10-s + (0.580 + 0.211i)13-s + (−0.939 + 0.342i)16-s + (0.473 + 0.397i)17-s − 18-s − 1.61·20-s + (−1.52 − 0.553i)25-s + (−0.309 − 0.535i)26-s + (−0.473 + 0.397i)29-s + (0.939 + 0.342i)32-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.280 + 1.59i)5-s + (0.500 − 0.866i)8-s + (0.766 − 0.642i)9-s + (1.23 − 1.04i)10-s + (0.580 + 0.211i)13-s + (−0.939 + 0.342i)16-s + (0.473 + 0.397i)17-s − 18-s − 1.61·20-s + (−1.52 − 0.553i)25-s + (−0.309 − 0.535i)26-s + (−0.473 + 0.397i)29-s + (0.939 + 0.342i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7614743099\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7614743099\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 5 | \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.580 - 0.211i)T + (0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (0.473 - 0.397i)T + (0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.61T + T^{2} \) |
| 41 | \( 1 + (1.52 - 0.553i)T + (0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.280 - 1.59i)T + (-0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.107 - 0.608i)T + (-0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (-1.52 + 0.553i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (1.52 + 0.553i)T + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (0.473 + 0.397i)T + (0.173 + 0.984i)T^{2} \) |
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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
−9.910574134290348824062945989396, −9.234528617189060214780303107775, −8.160506381989458275847102706197, −7.44006570502743885184519703207, −6.77000062238611348221767041858, −6.08752356484975135510317400569, −4.28673151630723036090616400514, −3.52607964610249815229108878008, −2.79088688570647037966452049275, −1.48008503117779649418788219783,
0.915433016892156785637691307476, 1.97397212393113611542562201465, 3.90299949164734803468449818062, 4.93436421257032907444698566167, 5.36791868070987748380259866184, 6.48232680570105308973536996848, 7.49678161397321880275168264672, 8.142090695641408242371078015642, 8.649982208160732320064676592225, 9.589196380418171138687093339885