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 + (0.323 − 1.48i)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 + (0.0562 − 0.257i)33-s + (0.816 − 0.816i)37-s + 0.260·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.373290110\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.373290110\) |
\(L(\frac{3}{2})\) |
|
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
−9.422921238612726626224994526585, −8.912791760668481940371282434589, −8.190136224159582784529525700727, −7.17636293198180912854310485021, −6.19070731289006391600214773565, −5.46700004930107318478413629937, −4.34674496586969934096368602194, −3.34656495077392585144628936113, −2.35521728604292314290350220223, −0.58082406430607239866596985523,
1.69271147179541110006609244539, 2.48302379069374012227227053807, 4.18396287574279451974598047570, 4.57730306573068018537114096775, 5.89666699301639009314891451740, 6.77421484596655795380562266592, 7.61300378212449512328718707479, 8.407500775644308947417947073649, 9.026237277565972267745147069174, 10.24891154124947401850312772616