L(s) = 1 | + (1.22 − 0.707i)2-s + (−0.991 − 0.130i)3-s + (0.499 − 0.866i)4-s + (−1.30 + 0.541i)6-s + (0.965 + 0.258i)9-s + (−0.608 + 0.793i)12-s + 1.84i·13-s + (0.499 + 0.866i)16-s + (1.36 − 0.366i)18-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)25-s + (1.30 + 2.26i)26-s + (−0.923 − 0.382i)27-s + (−0.662 − 0.382i)31-s + (1.22 + 0.707i)32-s + ⋯ |
L(s) = 1 | + (1.22 − 0.707i)2-s + (−0.991 − 0.130i)3-s + (0.499 − 0.866i)4-s + (−1.30 + 0.541i)6-s + (0.965 + 0.258i)9-s + (−0.608 + 0.793i)12-s + 1.84i·13-s + (0.499 + 0.866i)16-s + (1.36 − 0.366i)18-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)25-s + (1.30 + 2.26i)26-s + (−0.923 − 0.382i)27-s + (−0.662 − 0.382i)31-s + (1.22 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.636404356\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.636404356\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.991 + 0.130i)T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
good | 2 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.84iT - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 1.84T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.092892110343950804837524610926, −7.80221442025130754581165880205, −7.15252967957298059979243818270, −6.10363939871068646718476897917, −5.83337797633295750807238384544, −4.77441716542155369038288648369, −4.27381138830869096788982411750, −3.59187664014921901331979907160, −2.23410828267399492715557776456, −1.55096939180380605720388943851,
0.75980792629473654598607624957, 2.54219465530080986912465335385, 3.69722498688599629560276678257, 4.27692740533461123633177811090, 5.18882661549411043796001356486, 5.72415120929605171919023079588, 6.14746469249632844903841498149, 7.09039024326682596256118968179, 7.65051648431701365751788858375, 8.517555934835224971893358881142