L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.690 − 2.12i)3-s + (0.309 + 0.951i)4-s + (1.30 − 0.951i)5-s + (−1.80 + 1.31i)6-s + (−1.19 − 3.66i)7-s + (0.309 − 0.951i)8-s + (−1.61 − 1.17i)9-s − 1.61·10-s + (−3.04 + 1.31i)11-s + 2.23·12-s + (0.5 + 0.363i)13-s + (−1.19 + 3.66i)14-s + (−1.11 − 3.44i)15-s + (−0.809 + 0.587i)16-s + (1.42 − 1.03i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.398 − 1.22i)3-s + (0.154 + 0.475i)4-s + (0.585 − 0.425i)5-s + (−0.738 + 0.536i)6-s + (−0.450 − 1.38i)7-s + (0.109 − 0.336i)8-s + (−0.539 − 0.391i)9-s − 0.511·10-s + (−0.918 + 0.396i)11-s + 0.645·12-s + (0.138 + 0.100i)13-s + (−0.318 + 0.979i)14-s + (−0.288 − 0.888i)15-s + (−0.202 + 0.146i)16-s + (0.346 − 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.243185 - 1.11030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.243185 - 1.11030i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (3.04 - 1.31i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.690 + 2.12i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-1.30 + 0.951i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (1.19 + 3.66i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.363i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.42 + 1.03i)T + (5.25 - 16.1i)T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + (0.472 + 1.45i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.19 + 1.59i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.354 + 1.08i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.19 - 6.74i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 + (-0.781 + 2.40i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.66 - 7.02i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.590 + 1.81i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.35 + 3.88i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + (-6.66 + 4.84i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.42 + 13.6i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (2.42 + 1.76i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-8.16 + 5.93i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + (-8.42 - 6.12i)T + (29.9 + 92.2i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.63213442274900609524661853266, −9.946747639053860453595040333860, −9.013723071943378529427288330985, −7.85848563299355928948342703724, −7.33963685664800012518430944479, −6.49298818026468966313307820695, −4.94688603928653644385442678822, −3.37062197010359174089942532270, −2.04663983274109719744727023191, −0.854791012166043315870259358163,
2.40865985519220425276816131503, 3.41891448035543341709797459629, 5.15054567338486562520086448513, 5.74671846119690708713383112615, 6.89252941607184658502267321408, 8.386690577171578155865124820456, 8.863444109406005128690707991754, 9.852469240485717282773187782248, 10.25811975071947240501171026237, 11.20892372071604621922391867918