L(s) = 1 | + 2-s + (−1.5 − 0.866i)3-s + 4-s − 3.46i·5-s + (−1.5 − 0.866i)6-s + 7-s + 8-s + (1.5 + 2.59i)9-s − 3.46i·10-s + 3.46i·11-s + (−1.5 − 0.866i)12-s + 1.73i·13-s + 14-s + (−2.99 + 5.19i)15-s + 16-s − 1.73i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.866 − 0.499i)3-s + 0.5·4-s − 1.54i·5-s + (−0.612 − 0.353i)6-s + 0.377·7-s + 0.353·8-s + (0.5 + 0.866i)9-s − 1.09i·10-s + 1.04i·11-s + (−0.433 − 0.249i)12-s + 0.480i·13-s + 0.267·14-s + (−0.774 + 1.34i)15-s + 0.250·16-s − 0.420i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09994 - 0.553375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09994 - 0.553375i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 23 | \( 1 - 5.19iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 8.66iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−13.15489901751515059955542535017, −12.38098109882178152909384304628, −11.86405326257156272124322486984, −10.58246001556314574441209511667, −9.118119754959359622108277338484, −7.76232100624168519610049007081, −6.53910247390834776155477036816, −5.07148813490284239032473666909, −4.58512735759355047775599264373, −1.64369780833473190775040709017,
2.99063705687272663952860070260, 4.36832954540764876987581180628, 5.99522944989182225075727631750, 6.54564424096198188129094551060, 8.112262278083392832402569165284, 10.11430042684651296706286539675, 10.82738199385273632155144207714, 11.43843241121292595298890662752, 12.61561510748546313828220353693, 13.93865406482178534730604431979