| 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
−10.52838566225161487966085720360, −9.666315561907652670363550433131, −8.747122560134846947382452063048, −8.031563125032195332069694299277, −6.93892307643493041211638336217, −6.10750931422260296497565758890, −5.36660791819723302090138795322, −4.22333671755731136566954811878, −2.90379806582862881081652373243, −1.97926463275592165519936332302,
0.835537525156124629741059187479, 2.19215353786829350359294181795, 3.20427831868013738321927199387, 4.38292862876907506593037439814, 5.80985558517612891417254315971, 6.50340754338478934264957601481, 7.55616756354274764741347818317, 7.82778732689200152846334149443, 9.168757345056326447032469892370, 10.29467308281224797638954611365
