L(s) = 1 | + (0.689 − 1.19i)2-s + (−1.44 − 2.49i)3-s + (0.0491 + 0.0850i)4-s + (0.402 − 0.697i)5-s − 3.97·6-s + (1.26 + 2.32i)7-s + 2.89·8-s + (−2.65 + 4.59i)9-s + (−0.555 − 0.962i)10-s + (2.63 + 4.56i)11-s + (0.141 − 0.245i)12-s + (3.64 + 0.0965i)14-s − 2.32·15-s + (1.89 − 3.28i)16-s + (0.280 + 0.485i)17-s + (3.65 + 6.33i)18-s + ⋯ |
L(s) = 1 | + (0.487 − 0.844i)2-s + (−0.831 − 1.44i)3-s + (0.0245 + 0.0425i)4-s + (0.180 − 0.312i)5-s − 1.62·6-s + (0.476 + 0.878i)7-s + 1.02·8-s + (−0.883 + 1.53i)9-s + (−0.175 − 0.304i)10-s + (0.794 + 1.37i)11-s + (0.0408 − 0.0707i)12-s + (0.974 + 0.0257i)14-s − 0.599·15-s + (0.474 − 0.821i)16-s + (0.0679 + 0.117i)17-s + (0.861 + 1.49i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0369 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0369 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.095847401\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.095847401\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.26 - 2.32i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.689 + 1.19i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.44 + 2.49i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.402 + 0.697i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.63 - 4.56i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.280 - 0.485i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.92 + 5.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.802 + 1.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.28T + 29T^{2} \) |
| 31 | \( 1 + (-1.73 - 3.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.620 + 1.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.927T + 41T^{2} \) |
| 43 | \( 1 - 4.44T + 43T^{2} \) |
| 47 | \( 1 + (1.92 - 3.32i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.72 + 4.72i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.49 + 9.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.65 - 6.32i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.67 + 6.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.31T + 71T^{2} \) |
| 73 | \( 1 + (2.50 + 4.33i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.68 - 9.84i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.81T + 83T^{2} \) |
| 89 | \( 1 + (-2.50 + 4.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.643851717138965057301565365129, −8.728073245519789795538369653244, −7.63449882671781926063243910574, −7.09392509658426082578486177386, −6.26127000944665375228884527800, −5.11922364005417714181053933382, −4.60014683826657312467493008523, −2.93005099339245455285473365775, −1.94152533151549519083854020283, −1.27071550391386665308527803279,
1.08717782787712010528316468883, 3.36621192831533750392050606454, 4.18833415511438216368664507748, 4.87136132343856878104332846570, 5.88649916818609263380150149792, 6.16192584251624470822597602189, 7.26810592198515805805090587861, 8.234373371949000369783259901388, 9.310006067136146533581396228043, 10.31504113533338935350659610394