L(s) = 1 | + 2.29i·3-s − i·5-s − 7-s − 2.25·9-s + 0.744i·11-s + 3.83i·13-s + 2.29·15-s + 7.38·17-s + 4.58i·19-s − 2.29i·21-s + 2.51·23-s − 25-s + 1.70i·27-s − 4.80i·29-s − 3.09·31-s + ⋯ |
L(s) = 1 | + 1.32i·3-s − 0.447i·5-s − 0.377·7-s − 0.751·9-s + 0.224i·11-s + 1.06i·13-s + 0.591·15-s + 1.79·17-s + 1.05i·19-s − 0.500i·21-s + 0.523·23-s − 0.200·25-s + 0.328i·27-s − 0.891i·29-s − 0.555·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.521963742\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.521963742\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2.29iT - 3T^{2} \) |
| 11 | \( 1 - 0.744iT - 11T^{2} \) |
| 13 | \( 1 - 3.83iT - 13T^{2} \) |
| 17 | \( 1 - 7.38T + 17T^{2} \) |
| 19 | \( 1 - 4.58iT - 19T^{2} \) |
| 23 | \( 1 - 2.51T + 23T^{2} \) |
| 29 | \( 1 + 4.80iT - 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 + 11.6iT - 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 - 11.0iT - 53T^{2} \) |
| 59 | \( 1 - 11.0iT - 59T^{2} \) |
| 61 | \( 1 - 8.51iT - 61T^{2} \) |
| 67 | \( 1 - 10.5iT - 67T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 + 8.51T + 73T^{2} \) |
| 79 | \( 1 + 8.36T + 79T^{2} \) |
| 83 | \( 1 - 4.58iT - 83T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 + 3.38T + 97T^{2} \) |
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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
−9.323489999163891306412226953544, −8.920425685634941199064089835107, −7.82141725941560861789743849305, −7.10957768405317831640662136473, −5.79834433809768748447374985591, −5.44457580676116779578155352011, −4.19014763308781626847662267030, −3.97812349003314518335649740095, −2.83285097433587061983130582085, −1.37355136144138289882015142686,
0.56563651343461645109214901649, 1.61587753810156607365718105453, 2.93491750803413999228462643034, 3.35969732343852794289586471881, 4.99943899652972576363103865919, 5.73434249684400743738535240624, 6.63878150391994228799180016125, 7.11118850038665922263449866734, 7.907974257995763630127988178205, 8.422737323195057052221147365247