L(s) = 1 | + 2-s + (−1.66 + 0.459i)3-s + 4-s + (0.567 + 0.327i)5-s + (−1.66 + 0.459i)6-s + (−2.37 − 1.17i)7-s + 8-s + (2.57 − 1.53i)9-s + (0.567 + 0.327i)10-s + (1.54 − 2.67i)11-s + (−1.66 + 0.459i)12-s + (3.50 + 0.844i)13-s + (−2.37 − 1.17i)14-s + (−1.09 − 0.286i)15-s + 16-s + 7.02·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.964 + 0.265i)3-s + 0.5·4-s + (0.253 + 0.146i)5-s + (−0.681 + 0.187i)6-s + (−0.896 − 0.442i)7-s + 0.353·8-s + (0.858 − 0.512i)9-s + (0.179 + 0.103i)10-s + (0.466 − 0.807i)11-s + (−0.482 + 0.132i)12-s + (0.972 + 0.234i)13-s + (−0.634 − 0.312i)14-s + (−0.283 − 0.0738i)15-s + 0.250·16-s + 1.70·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.74171 - 0.0362795i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.74171 - 0.0362795i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.66 - 0.459i)T \) |
| 7 | \( 1 + (2.37 + 1.17i)T \) |
| 13 | \( 1 + (-3.50 - 0.844i)T \) |
good | 5 | \( 1 + (-0.567 - 0.327i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.54 + 2.67i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 7.02T + 17T^{2} \) |
| 19 | \( 1 + (-3.25 - 5.64i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.43iT - 23T^{2} \) |
| 29 | \( 1 + (3.89 - 2.24i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.26 - 3.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.96iT - 37T^{2} \) |
| 41 | \( 1 + (-7.52 + 4.34i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0380 - 0.0658i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (8.04 + 4.64i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.68 + 5.59i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 7.07iT - 59T^{2} \) |
| 61 | \( 1 + (13.2 - 7.67i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 - 1.99i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.469 + 0.813i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.44 + 9.43i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.40 - 2.42i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.24iT - 83T^{2} \) |
| 89 | \( 1 - 10.9iT - 89T^{2} \) |
| 97 | \( 1 + (8.57 - 14.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.75278694887515701375338827974, −10.21817150744750762804151680755, −9.335815137221729745941511945351, −7.889302697045944027433277158487, −6.77041999441198461416470009873, −5.99733757143179811943994409664, −5.49144693653420540723420905563, −3.91911232769111729172311117249, −3.40473041359187511000137611759, −1.18097078425202904478797824715,
1.32890729275950350782142315334, 3.01543989489878090154286756241, 4.25485154211126771845127824706, 5.49361089238560663808354529225, 5.95029035280117928039759220658, 6.94619621253580287039926000110, 7.76877479407636329807858152470, 9.416969698547092357377028433081, 9.904326819609983080318985876628, 11.11686432396216884887318152373