L(s) = 1 | + (0.323 + 0.323i)3-s + (−0.901 − 0.901i)7-s − 2.79i·9-s + (−1.79 + 2.79i)11-s + (−2.51 + 2.51i)13-s + (5.22 + 5.22i)17-s + 5·19-s − 0.582i·21-s + (2.83 + 2.83i)23-s + (1.87 − 1.87i)27-s + 8.37·29-s + 4.58·31-s + (−1.48 + 0.323i)33-s + (4.96 − 4.96i)37-s − 1.62·39-s + ⋯ |
L(s) = 1 | + (0.186 + 0.186i)3-s + (−0.340 − 0.340i)7-s − 0.930i·9-s + (−0.540 + 0.841i)11-s + (−0.698 + 0.698i)13-s + (1.26 + 1.26i)17-s + 1.14·19-s − 0.127i·21-s + (0.592 + 0.592i)23-s + (0.360 − 0.360i)27-s + 1.55·29-s + 0.823·31-s + (−0.257 + 0.0562i)33-s + (0.816 − 0.816i)37-s − 0.260·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.635328935\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.635328935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (1.79 - 2.79i)T \) |
good | 3 | \( 1 + (-0.323 - 0.323i)T + 3iT^{2} \) |
| 7 | \( 1 + (0.901 + 0.901i)T + 7iT^{2} \) |
| 13 | \( 1 + (2.51 - 2.51i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5.22 - 5.22i)T + 17iT^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (-2.83 - 2.83i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.37T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + (-4.96 + 4.96i)T - 37iT^{2} \) |
| 41 | \( 1 + 5iT - 41T^{2} \) |
| 43 | \( 1 + (6.83 - 6.83i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.578 - 0.578i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.09 - 7.09i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.58iT - 59T^{2} \) |
| 61 | \( 1 - 7.79iT - 61T^{2} \) |
| 67 | \( 1 + (6.19 - 6.19i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.417T + 71T^{2} \) |
| 73 | \( 1 + (0.901 - 0.901i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.20T + 79T^{2} \) |
| 83 | \( 1 + (5.22 - 5.22i)T - 83iT^{2} \) |
| 89 | \( 1 - 4.79iT - 89T^{2} \) |
| 97 | \( 1 + (-3.35 + 3.35i)T - 97iT^{2} \) |
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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
−9.915617985417953743193785087796, −9.323119989708159339906337648490, −8.254382284867585044008875215352, −7.39810591241107100171663770751, −6.68286951291388117178968210085, −5.66920303469979311569479894703, −4.63690977939394535058905314651, −3.69230020363874268990680757965, −2.74849544593921317877954758797, −1.18110071039350545922711705183,
0.867601483230896735814702437595, 2.77599928629211002028491996623, 3.02763377765856675825723540275, 4.91781490046104132841559445726, 5.28850876691964703120027854930, 6.43080479709935565452880169689, 7.52979600611865278945682643290, 7.996779683301576404150993542776, 8.880177568251259320951975009109, 10.00058461472580234417997629462