L(s) = 1 | + (0.933 − 1.17i)2-s + (−1.67 − 0.808i)3-s + (−0.0535 − 0.234i)4-s + (2.90 + 1.39i)5-s + (−2.51 + 1.21i)6-s + (−2.13 + 1.56i)7-s + (2.37 + 1.14i)8-s + (0.293 + 0.368i)9-s + (4.35 − 2.09i)10-s + (−2.84 + 3.56i)11-s + (−0.0997 + 0.436i)12-s + (0.623 − 0.781i)13-s + (−0.162 + 3.95i)14-s + (−3.74 − 4.69i)15-s + (3.98 − 1.91i)16-s + (−1.67 + 7.35i)17-s + ⋯ |
L(s) = 1 | + (0.659 − 0.827i)2-s + (−0.969 − 0.466i)3-s + (−0.0267 − 0.117i)4-s + (1.30 + 0.626i)5-s + (−1.02 + 0.494i)6-s + (−0.806 + 0.590i)7-s + (0.838 + 0.404i)8-s + (0.0979 + 0.122i)9-s + (1.37 − 0.662i)10-s + (−0.856 + 1.07i)11-s + (−0.0287 + 0.126i)12-s + (0.172 − 0.216i)13-s + (−0.0433 + 1.05i)14-s + (−0.967 − 1.21i)15-s + (0.996 − 0.479i)16-s + (−0.407 + 1.78i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.66074 + 0.199036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66074 + 0.199036i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.13 - 1.56i)T \) |
| 13 | \( 1 + (-0.623 + 0.781i)T \) |
good | 2 | \( 1 + (-0.933 + 1.17i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (1.67 + 0.808i)T + (1.87 + 2.34i)T^{2} \) |
| 5 | \( 1 + (-2.90 - 1.39i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (2.84 - 3.56i)T + (-2.44 - 10.7i)T^{2} \) |
| 17 | \( 1 + (1.67 - 7.35i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 23 | \( 1 + (-1.22 - 5.36i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-1.76 + 7.71i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 + (-1.96 + 8.61i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (-4.09 - 1.97i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-7.13 + 3.43i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.663 + 0.831i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (2.34 + 10.2i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (3.16 - 1.52i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (1.26 - 5.56i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 + (-1.51 - 6.65i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-2.88 - 3.62i)T + (-16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + (1.25 + 1.57i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-8.37 - 10.5i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 0.268T + 97T^{2} \) |
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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
−10.76322880269786353958152537270, −10.07485195360691148384441554634, −9.285947741242050560877646253613, −7.72055416827124506383537864843, −6.79849336596893284005279880477, −5.71657170936091764752753107136, −5.51377752333062838399332811954, −3.84698687889268686908242035294, −2.63013246097441834811197809326, −1.81219114651911382104373056608,
0.843303714241304073017402314067, 2.94231795585249877130071769702, 4.64963393184240850544320611159, 5.20461805926689023046819448464, 5.86562563460144722536341404235, 6.55048812898887139069186200195, 7.53625152631326343790142334323, 9.028392373221124979886778108697, 9.748190149953228939837849229857, 10.61778412626806992276842239830