L(s) = 1 | − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−1.18 − 2.05i)7-s − 8-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−2.5 + 4.33i)11-s + (−0.5 − 0.866i)12-s + (−1 + 1.73i)13-s + (1.18 + 2.05i)14-s − 0.999·15-s + 16-s + (2.68 + 4.65i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.448 − 0.776i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.753 + 1.30i)11-s + (−0.144 − 0.249i)12-s + (−0.277 + 0.480i)13-s + (0.317 + 0.549i)14-s − 0.258·15-s + 0.250·16-s + (0.651 + 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.853653 + 0.147936i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.853653 + 0.147936i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-5.55 - 0.322i)T \) |
good | 7 | \( 1 + (1.18 + 2.05i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.68 - 4.65i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.37 - 4.10i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 8.11T + 23T^{2} \) |
| 29 | \( 1 + 3.62T + 29T^{2} \) |
| 37 | \( 1 + (5.05 + 8.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.37 - 5.84i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.68 + 2.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + (-0.813 + 1.40i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.55 - 9.62i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (-5.05 + 8.76i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.37 + 2.37i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.68 - 2.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.18 - 2.05i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−10.19716702708871181340521406130, −9.376765932094052864187408972155, −8.393920109924620139091001081144, −7.39089181393121718319126531448, −7.09079789385031630542416818889, −5.92130816100626113549252165703, −4.98625780949143987480201255430, −3.71838102405561058449118389807, −2.23409067287798234259356776695, −1.11750184894846967522022142589,
0.63386092144992984708533358353, 2.82837234325315307458534544311, 3.10398348369223538621907283531, 5.11005805207599417458304080231, 5.59748266699821145733577534922, 6.66211097316113823045530139873, 7.51584154488483455990595000267, 8.616182750693696897321653373661, 9.202577115439549784354415157490, 9.998474093833176682499264269001