L(s) = 1 | − i·2-s − 4-s − 0.717·5-s + i·8-s + 0.717i·10-s − 3i·11-s + 2.44i·13-s + 16-s − 5.91·17-s − 5.91i·19-s + 0.717·20-s − 3·22-s − 4.24i·23-s − 4.48·25-s + 2.44·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.320·5-s + 0.353i·8-s + 0.226i·10-s − 0.904i·11-s + 0.679i·13-s + 0.250·16-s − 1.43·17-s − 1.35i·19-s + 0.160·20-s − 0.639·22-s − 0.884i·23-s − 0.897·25-s + 0.480·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0280346 + 0.570252i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0280346 + 0.570252i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.717T + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 + 5.91iT - 19T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 7.24iT - 29T^{2} \) |
| 31 | \( 1 - 9.08iT - 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 0.242T + 43T^{2} \) |
| 47 | \( 1 + 5.91T + 47T^{2} \) |
| 53 | \( 1 + 7.24iT - 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 - 1.01iT - 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 1.75iT - 71T^{2} \) |
| 73 | \( 1 - 1.43iT - 73T^{2} \) |
| 79 | \( 1 + 2.75T + 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−9.790522803334570655112082959601, −8.739168696634809321022170416047, −8.458813720151161865112782899079, −7.05472489745851859677858019540, −6.30876961478634493065263286809, −4.98268882295348862251349393880, −4.21556245666810672954805328143, −3.11767502456994874850778429874, −2.00552691845123296852125115724, −0.26326806086560037148294724131,
1.85586136600495704495656084185, 3.48995157079225962609680314174, 4.41685230965194117624707237538, 5.39525076564707989804185063717, 6.30662964856177018954704914913, 7.26636157553289107575935041670, 7.895662314751417095197825682938, 8.750456647455199800393788002497, 9.690895613793729834012088731491, 10.35592227203560011496717977173