L(s) = 1 | + 1.36i·2-s − i·3-s + 0.141·4-s + 1.36·6-s + 2.50i·7-s + 2.91i·8-s − 9-s + 11-s − 0.141i·12-s − 1.14i·13-s − 3.41·14-s − 3.69·16-s + 7.64i·17-s − 1.36i·18-s − 1.77·19-s + ⋯ |
L(s) = 1 | + 0.964i·2-s − 0.577i·3-s + 0.0706·4-s + 0.556·6-s + 0.946i·7-s + 1.03i·8-s − 0.333·9-s + 0.301·11-s − 0.0408i·12-s − 0.316i·13-s − 0.912·14-s − 0.924·16-s + 1.85i·17-s − 0.321i·18-s − 0.407·19-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
\(L(1)\) |
\(\approx\) |
\(0.840395 + 1.35978i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.840395 + 1.35978i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.36iT - 2T^{2} \) |
| 7 | \( 1 - 2.50iT - 7T^{2} \) |
| 13 | \( 1 + 1.14iT - 13T^{2} \) |
| 17 | \( 1 - 7.64iT - 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 0.726T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 + 8.42iT - 37T^{2} \) |
| 41 | \( 1 - 0.636T + 41T^{2} \) |
| 43 | \( 1 - 12.6iT - 43T^{2} \) |
| 47 | \( 1 - 6.14iT - 47T^{2} \) |
| 53 | \( 1 - 12.0iT - 53T^{2} \) |
| 59 | \( 1 - 3.41T + 59T^{2} \) |
| 61 | \( 1 - 4.59T + 61T^{2} \) |
| 67 | \( 1 + 9.32iT - 67T^{2} \) |
| 71 | \( 1 - 5.85T + 71T^{2} \) |
| 73 | \( 1 + 7.55iT - 73T^{2} \) |
| 79 | \( 1 + 6.91T + 79T^{2} \) |
| 83 | \( 1 - 6.17iT - 83T^{2} \) |
| 89 | \( 1 + 3.45T + 89T^{2} \) |
| 97 | \( 1 + 19.4iT - 97T^{2} \) |
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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
−10.65004035351097424352994618335, −9.332481895112223114183513356203, −8.416444927760385272075907488360, −8.012379542549033958867909808643, −6.96359711452926003331953126943, −6.04322377474091933811099576631, −5.79122601932226475460322227704, −4.40477711544612496636891062686, −2.83294617430316222245350672914, −1.76338113636392119566882099693,
0.789668403743869964662877748110, 2.32648292819594252770638668375, 3.42886479257348325944268052734, 4.21327859517911766577734547427, 5.21840213223943142923554578482, 6.70949637662139867228051969005, 7.17647913187698786037813900902, 8.494554817135161229854542717659, 9.526755138888262585547647389072, 10.06147372378910565847734877043