L(s) = 1 | + 2-s + 4-s + (0.5 − 0.866i)5-s + (−2 − 3.46i)7-s + 8-s + (1.5 + 2.59i)9-s + (0.5 − 0.866i)10-s + 5·11-s + (−1 − 1.73i)13-s + (−2 − 3.46i)14-s + 16-s + (−1.5 − 2.59i)17-s + (1.5 + 2.59i)18-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.223 − 0.387i)5-s + (−0.755 − 1.30i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s + (0.158 − 0.273i)10-s + 1.50·11-s + (−0.277 − 0.480i)13-s + (−0.534 − 0.925i)14-s + 0.250·16-s + (−0.363 − 0.630i)17-s + (0.353 + 0.612i)18-s + (−0.114 + 0.198i)19-s + (0.111 − 0.193i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07322 - 0.652502i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07322 - 0.652502i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (2.5 - 6.06i)T \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4 - 6.92i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−11.01653457444927827685535149302, −10.29350549504631398542209081318, −9.467158761602729535618337261593, −8.215379947118403235266434338776, −6.95276715117128113809300309244, −6.60880205782144486275819730425, −5.03496403415619514941170843025, −4.27985884724954023545862541166, −3.15211712236260854265861609405, −1.34299272733649711170949435391,
1.94743167624110913464808553555, 3.30017773388887357266373240934, 4.23132523229203650692170866467, 5.75071674329795722233131610543, 6.42596188995669974106729873502, 7.08385977116081676069743870649, 8.849824913254579871728753888944, 9.332356455626110377096081100883, 10.33549292856202067221691005970, 11.76558643320634296254846706045