| L(s) = 1 | − i·2-s − 4-s + i·5-s + 3.43·7-s + i·8-s + 10-s + 1.45·11-s − 4.73i·13-s − 3.43i·14-s + 16-s − 7.23·17-s − 5.86·19-s − i·20-s − 1.45i·22-s + 0.129·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 1.30·7-s + 0.353i·8-s + 0.316·10-s + 0.438·11-s − 1.31i·13-s − 0.919i·14-s + 0.250·16-s − 1.75·17-s − 1.34·19-s − 0.223i·20-s − 0.309i·22-s + 0.0270·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.301191418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301191418\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 31 | \( 1 + (0.289 + 5.56i)T \) |
| good | 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 4.73iT - 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 19 | \( 1 + 5.86T + 19T^{2} \) |
| 23 | \( 1 - 0.129T + 23T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 37 | \( 1 + 2.40iT - 37T^{2} \) |
| 41 | \( 1 + 11.1iT - 41T^{2} \) |
| 43 | \( 1 + 9.60iT - 43T^{2} \) |
| 47 | \( 1 - 5.46iT - 47T^{2} \) |
| 53 | \( 1 - 8.40T + 53T^{2} \) |
| 59 | \( 1 - 1.00iT - 59T^{2} \) |
| 61 | \( 1 - 1.15iT - 61T^{2} \) |
| 67 | \( 1 - 1.00T + 67T^{2} \) |
| 71 | \( 1 - 6.84iT - 71T^{2} \) |
| 73 | \( 1 + 14.1iT - 73T^{2} \) |
| 79 | \( 1 - 1.39iT - 79T^{2} \) |
| 83 | \( 1 + 0.628T + 83T^{2} \) |
| 89 | \( 1 - 6.70T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−8.665916458872671517410508089743, −7.88121142870250347153041636170, −7.11241358081305587152838926113, −6.10264119405841582257346898130, −5.28106378648393047693944100673, −4.37478575495388144810212641456, −3.78121153826471669445377586683, −2.44389329319335167173439615350, −1.92658918527005668711367316211, −0.41109918612795852094923154415,
1.40688970359446308461631801642, 2.25159115959386425113638173544, 3.96970903367319807223052763684, 4.57874421103104976813790901014, 5.00371031919607863542756992120, 6.33011282901787272754143060371, 6.65565229341610030568049274800, 7.63063449478998367580726249337, 8.545422255514463696583794966892, 8.738520732505389852022792426298
