L(s) = 1 | + 2.28·2-s + (−0.5 − 0.866i)3-s + 3.19·4-s + (−1.46 − 2.54i)5-s + (−1.14 − 1.97i)6-s + (0.102 − 2.64i)7-s + 2.73·8-s + (−0.499 + 0.866i)9-s + (−3.34 − 5.80i)10-s + (2.58 + 4.47i)11-s + (−1.59 − 2.77i)12-s + (−0.364 + 3.58i)13-s + (0.233 − 6.02i)14-s + (−1.46 + 2.54i)15-s − 0.160·16-s + 5.05·17-s + ⋯ |
L(s) = 1 | + 1.61·2-s + (−0.288 − 0.499i)3-s + 1.59·4-s + (−0.656 − 1.13i)5-s + (−0.465 − 0.806i)6-s + (0.0386 − 0.999i)7-s + 0.967·8-s + (−0.166 + 0.288i)9-s + (−1.05 − 1.83i)10-s + (0.779 + 1.34i)11-s + (−0.461 − 0.799i)12-s + (−0.101 + 0.994i)13-s + (0.0623 − 1.61i)14-s + (−0.379 + 0.656i)15-s − 0.0400·16-s + 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.18030 - 1.18436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18030 - 1.18436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.102 + 2.64i)T \) |
| 13 | \( 1 + (0.364 - 3.58i)T \) |
good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 5 | \( 1 + (1.46 + 2.54i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.58 - 4.47i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + (1.12 - 1.95i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 + (-0.216 + 0.375i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.34 - 2.32i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + (-0.269 + 0.466i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.66 + 8.07i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.87 + 8.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.377 + 0.653i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 3.64T + 59T^{2} \) |
| 61 | \( 1 + (3.47 - 6.02i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.68 - 11.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.90 - 6.76i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.94 - 13.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.79 + 13.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 4.00T + 89T^{2} \) |
| 97 | \( 1 + (-2.69 - 4.67i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.10530705210719040306685249127, −11.47988121930211731489894085652, −10.07983150421954703304953117272, −8.728939623067913034067439447698, −7.30649277633843969704429201218, −6.79342043487969818190329462438, −5.28934108780019816300840507278, −4.49238899200855099363458676812, −3.72190375772915885587229268938, −1.56126757544239471318631709952,
3.13555820901389203541016819044, 3.31140859853744467556700626122, 4.91404241155007593339049277541, 5.86396888561439277115209614233, 6.57141592815137390973764138375, 7.953203376509852387561018196478, 9.280617373684464197145293229935, 10.80534568491051517220564334306, 11.28527410141985051036729837091, 12.03713868192152448145227892417