L(s) = 1 | − 0.414i·2-s − i·3-s + 1.82·4-s − 0.414·6-s − 2.41i·7-s − 1.58i·8-s − 9-s − 11-s − 1.82i·12-s − 2.82i·13-s − 0.999·14-s + 3·16-s − 0.414i·17-s + 0.414i·18-s − 3.58·19-s + ⋯ |
L(s) = 1 | − 0.292i·2-s − 0.577i·3-s + 0.914·4-s − 0.169·6-s − 0.912i·7-s − 0.560i·8-s − 0.333·9-s − 0.301·11-s − 0.527i·12-s − 0.784i·13-s − 0.267·14-s + 0.750·16-s − 0.100i·17-s + 0.0976i·18-s − 0.822·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.918372 - 1.48595i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.918372 - 1.48595i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 0.414iT - 2T^{2} \) |
| 7 | \( 1 + 2.41iT - 7T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 0.414iT - 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 5.82iT - 37T^{2} \) |
| 41 | \( 1 - 8.89T + 41T^{2} \) |
| 43 | \( 1 - 0.343iT - 43T^{2} \) |
| 47 | \( 1 - 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 3.65iT - 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 + 11.6iT - 67T^{2} \) |
| 71 | \( 1 - 2.17T + 71T^{2} \) |
| 73 | \( 1 - 3.17iT - 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 - 0.171iT - 97T^{2} \) |
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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
−10.29467308281224797638954611365, −9.168757345056326447032469892370, −7.82778732689200152846334149443, −7.55616756354274764741347818317, −6.50340754338478934264957601481, −5.80985558517612891417254315971, −4.38292862876907506593037439814, −3.20427831868013738321927199387, −2.19215353786829350359294181795, −0.835537525156124629741059187479,
1.97926463275592165519936332302, 2.90379806582862881081652373243, 4.22333671755731136566954811878, 5.36660791819723302090138795322, 6.10750931422260296497565758890, 6.93892307643493041211638336217, 8.031563125032195332069694299277, 8.747122560134846947382452063048, 9.666315561907652670363550433131, 10.52838566225161487966085720360