L(s) = 1 | + (1.40 + 0.183i)2-s + (−1.57 + 0.711i)3-s + (1.93 + 0.514i)4-s + 2.07·5-s + (−2.34 + 0.708i)6-s − 4.67i·7-s + (2.61 + 1.07i)8-s + (1.98 − 2.24i)9-s + (2.90 + 0.380i)10-s + 0.907i·11-s + (−3.41 + 0.562i)12-s − 0.234i·13-s + (0.858 − 6.56i)14-s + (−3.27 + 1.47i)15-s + (3.47 + 1.98i)16-s + 1.17i·17-s + ⋯ |
L(s) = 1 | + (0.991 + 0.129i)2-s + (−0.911 + 0.410i)3-s + (0.966 + 0.257i)4-s + 0.926·5-s + (−0.957 + 0.289i)6-s − 1.76i·7-s + (0.924 + 0.380i)8-s + (0.662 − 0.749i)9-s + (0.918 + 0.120i)10-s + 0.273i·11-s + (−0.986 + 0.162i)12-s − 0.0649i·13-s + (0.229 − 1.75i)14-s + (−0.844 + 0.380i)15-s + (0.867 + 0.497i)16-s + 0.284i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.30179 + 0.0380702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.30179 + 0.0380702i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.183i)T \) |
| 3 | \( 1 + (1.57 - 0.711i)T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 2.07T + 5T^{2} \) |
| 7 | \( 1 + 4.67iT - 7T^{2} \) |
| 11 | \( 1 - 0.907iT - 11T^{2} \) |
| 13 | \( 1 + 0.234iT - 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 23 | \( 1 + 0.116T + 23T^{2} \) |
| 29 | \( 1 - 7.50T + 29T^{2} \) |
| 31 | \( 1 - 4.34iT - 31T^{2} \) |
| 37 | \( 1 - 10.2iT - 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 - 0.108T + 53T^{2} \) |
| 59 | \( 1 + 12.9iT - 59T^{2} \) |
| 61 | \( 1 + 2.38iT - 61T^{2} \) |
| 67 | \( 1 - 2.38T + 67T^{2} \) |
| 71 | \( 1 - 2.06T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 9.23iT - 79T^{2} \) |
| 83 | \( 1 - 8.40iT - 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 + 6.29T + 97T^{2} \) |
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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
−11.03413501982320184053652252000, −10.28860952861030914992908183935, −9.891220674359328825391262132137, −8.065457464286510981506658865732, −6.77461335174813752663058362861, −6.49072350271092181905543334398, −5.18253433482952957579961484894, −4.47531363420354161216215198725, −3.42418859517802802983121818943, −1.45633033675159975999806574196,
1.81421695605559869092568793600, 2.74077593961375757061215782409, 4.62061021827506775897581955230, 5.65252211624456695859994630670, 5.93070871024334779500140728498, 6.87698040994389779755530489628, 8.246767391919926284951056657247, 9.503637150041145533691510613443, 10.39379493676112613854352460491, 11.47751058816026812313702625846