L(s) = 1 | + (−0.988 − 0.149i)3-s + (0.0747 + 0.997i)4-s + (0.955 + 0.294i)9-s + (0.0747 − 0.997i)12-s + (−0.277 + 1.21i)13-s + (−0.988 + 0.149i)16-s + (0.222 − 0.385i)19-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (0.900 + 1.56i)31-s + (−0.222 + 0.974i)36-s + (−0.134 + 1.79i)37-s + (0.455 − 1.16i)39-s + (−0.277 − 0.347i)43-s + 0.999·48-s + ⋯ |
L(s) = 1 | + (−0.988 − 0.149i)3-s + (0.0747 + 0.997i)4-s + (0.955 + 0.294i)9-s + (0.0747 − 0.997i)12-s + (−0.277 + 1.21i)13-s + (−0.988 + 0.149i)16-s + (0.222 − 0.385i)19-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (0.900 + 1.56i)31-s + (−0.222 + 0.974i)36-s + (−0.134 + 1.79i)37-s + (0.455 − 1.16i)39-s + (−0.277 − 0.347i)43-s + 0.999·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0434 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0434 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6719669046\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6719669046\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.988 + 0.149i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 5 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 11 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 13 | \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 19 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.134 - 1.79i)T + (-0.988 - 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 53 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 61 | \( 1 + (0.134 - 1.79i)T + (-0.988 - 0.149i)T^{2} \) |
| 67 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-1.32 + 1.22i)T + (0.0747 - 0.997i)T^{2} \) |
| 79 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−10.47518104424875528965449279365, −9.535671400754801822113764356716, −8.678549467132518953664434876158, −7.67734276494445089857634703513, −6.93416687810977633319096877358, −6.33642319891838762922884936474, −5.02419548136428009721815165633, −4.35395177572868319988591433312, −3.18612379787838643818092983716, −1.75156317040646624121906754612,
0.72629210333105241839107935149, 2.24976659757639213257908347869, 3.90798508218781703799458124185, 4.93582195642617589152624198617, 5.70720278963695703247105299143, 6.22643050299288355965122117062, 7.27584823894648976497076982624, 8.180053457758333693130705472062, 9.596697516362718844040107937922, 9.914363126508631061329429082679