| L(s) = 1 | − i·2-s + i·3-s − 4-s + 4.30i·5-s + 6-s + i·8-s − 9-s + 4.30·10-s + (2.11 − 2.55i)11-s − i·12-s − 1.00·13-s − 4.30·15-s + 16-s − 3.33·17-s + i·18-s + 3.23·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 1.92i·5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + 1.36·10-s + (0.637 − 0.770i)11-s − 0.288i·12-s − 0.277·13-s − 1.11·15-s + 0.250·16-s − 0.809·17-s + 0.235i·18-s + 0.741·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4430454025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4430454025\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-2.11 + 2.55i)T \) |
| good | 5 | \( 1 - 4.30iT - 5T^{2} \) |
| 13 | \( 1 + 1.00T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 - 7.74iT - 29T^{2} \) |
| 31 | \( 1 - 3.20iT - 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 + 4.14iT - 43T^{2} \) |
| 47 | \( 1 - 3.20iT - 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 3.44iT - 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 + 0.330T + 67T^{2} \) |
| 71 | \( 1 - 2.84T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 16.0iT - 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 + 2.78iT - 89T^{2} \) |
| 97 | \( 1 - 12.3iT - 97T^{2} \) |
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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
−9.248438937656856097734500433250, −8.464997450528197412044555139725, −7.51741171874341842309176634850, −6.73825923867954398039754841659, −6.10198056011674622816998132090, −5.17508875079386657186547981661, −4.02758826199536295216260381986, −3.39888594248925660248859554807, −2.83287037034891801650749589102, −1.79500633690044452747113684719,
0.13684830930995699625971290472, 1.28106411720548747693110305469, 2.19320146268672705051973130308, 3.96936145908651493709577283897, 4.46909130725198763615131180320, 5.28160298441492412816189744497, 5.95863459155089554027235994313, 6.77410346508066632480028580421, 7.68894120420085916775679456492, 8.173117058558663580884992365022
