L(s) = 1 | + (0.0646 − 0.198i)5-s + (1.47 − 1.07i)7-s + (−0.309 − 0.951i)9-s + (−0.913 + 0.406i)11-s + (−0.395 − 0.128i)17-s + (−0.809 − 0.587i)19-s − 1.90i·23-s + (0.773 + 0.562i)25-s + (−0.118 − 0.363i)35-s − 1.33·43-s − 0.209·45-s + (−0.873 + 1.20i)47-s + (0.722 − 2.22i)49-s + (0.0218 + 0.207i)55-s + (1.41 + 0.459i)61-s + ⋯ |
L(s) = 1 | + (0.0646 − 0.198i)5-s + (1.47 − 1.07i)7-s + (−0.309 − 0.951i)9-s + (−0.913 + 0.406i)11-s + (−0.395 − 0.128i)17-s + (−0.809 − 0.587i)19-s − 1.90i·23-s + (0.773 + 0.562i)25-s + (−0.118 − 0.363i)35-s − 1.33·43-s − 0.209·45-s + (−0.873 + 1.20i)47-s + (0.722 − 2.22i)49-s + (0.0218 + 0.207i)55-s + (1.41 + 0.459i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.248518733\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.248518733\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (0.913 - 0.406i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
good | 3 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.0646 + 0.198i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-1.47 + 1.07i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.395 + 0.128i)T + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + 1.90iT - T^{2} \) |
| 29 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + 1.33T + T^{2} \) |
| 47 | \( 1 + (0.873 - 1.20i)T + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.41 - 0.459i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-1.01 - 1.40i)T + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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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
−8.390103939119448802946486188222, −8.116176942749872675291946276506, −6.99630713709846989180082094979, −6.65019193582034936331314850522, −5.39851658735085209214933087621, −4.66130301722823186653502808656, −4.22838478272006408838656657285, −2.97530578652966748175603832613, −1.94913409745606696139706345422, −0.72980679512534452326994036295,
1.77974243532811677404699898816, 2.29426301022963154966370326109, 3.36044853661878431204484434096, 4.63851267910986173013885020735, 5.27393514791810203439186107005, 5.67712064964245757522246252142, 6.76585706916765646790548685378, 7.83618544586968431039956619739, 8.221228232817300886287606872788, 8.665657554051697098127700303354