L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.365 + 1.69i)3-s + (−0.499 + 0.866i)4-s + (0.231 − 0.400i)5-s + (1.28 − 1.16i)6-s + (0.165 + 0.286i)7-s + 0.999·8-s + (−2.73 + 1.23i)9-s − 0.462·10-s + (2.28 + 3.95i)11-s + (−1.64 − 0.529i)12-s + (−0.380 + 0.658i)13-s + (0.165 − 0.286i)14-s + (0.762 + 0.244i)15-s + (−0.5 − 0.866i)16-s + 5.46·17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.211 + 0.977i)3-s + (−0.249 + 0.433i)4-s + (0.103 − 0.179i)5-s + (0.523 − 0.474i)6-s + (0.0625 + 0.108i)7-s + 0.353·8-s + (−0.910 + 0.412i)9-s − 0.146·10-s + (0.688 + 1.19i)11-s + (−0.476 − 0.152i)12-s + (−0.105 + 0.182i)13-s + (0.0442 − 0.0765i)14-s + (0.196 + 0.0632i)15-s + (−0.125 − 0.216i)16-s + 1.32·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.961050 + 0.605011i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.961050 + 0.605011i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.365 - 1.69i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.231 + 0.400i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.165 - 0.286i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.28 - 3.95i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.380 - 0.658i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 + 8.02T + 19T^{2} \) |
| 29 | \( 1 + (-3.24 - 5.62i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.49 - 7.78i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.26T + 37T^{2} \) |
| 41 | \( 1 + (-3.11 + 5.39i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.391 + 0.678i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.79 - 6.57i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 + (-6.14 + 10.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.55 + 9.61i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.83 + 11.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.24T + 71T^{2} \) |
| 73 | \( 1 + 9.95T + 73T^{2} \) |
| 79 | \( 1 + (-1.76 - 3.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.95 - 6.84i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 1.01T + 89T^{2} \) |
| 97 | \( 1 + (1.08 + 1.87i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.08599172818384657996699169661, −10.46136873611879185071505388490, −9.552064106776006776484100895139, −8.999539146303558661299187162241, −8.027239418471244710072734230373, −6.78278690812690403602266349352, −5.27783613229160075058067771684, −4.35849565191758322227537266352, −3.32729291849535757563414323756, −1.86138002185800806279822413499,
0.849478866672950339283296662898, 2.56307929554595840624920767424, 4.10345573663123232588973616543, 5.92739530617229548939175162145, 6.22240907301420646198189635758, 7.42795376627144161828966309835, 8.233195105577668973520444773597, 8.877997507602535848553749112059, 10.05611101729119139524425770311, 11.07085218237249561772021048659