L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (2.13 − 0.693i)5-s + (−0.309 − 0.951i)6-s + (1.63 − 2.08i)7-s + (−0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s − 2.24·10-s + (−1.12 − 3.11i)11-s + 0.999i·12-s + (1.40 − 4.33i)13-s + (−2.19 + 1.47i)14-s + (1.81 + 1.31i)15-s + (0.309 + 0.951i)16-s + (0.242 + 0.746i)17-s + ⋯ |
L(s) = 1 | + (−0.672 − 0.218i)2-s + (0.339 + 0.467i)3-s + (0.404 + 0.293i)4-s + (0.954 − 0.310i)5-s + (−0.126 − 0.388i)6-s + (0.617 − 0.786i)7-s + (−0.207 − 0.286i)8-s + (−0.103 + 0.317i)9-s − 0.709·10-s + (−0.339 − 0.940i)11-s + 0.288i·12-s + (0.390 − 1.20i)13-s + (−0.586 + 0.394i)14-s + (0.468 + 0.340i)15-s + (0.0772 + 0.237i)16-s + (0.0588 + 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32028 - 0.448436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32028 - 0.448436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 7 | \( 1 + (-1.63 + 2.08i)T \) |
| 11 | \( 1 + (1.12 + 3.11i)T \) |
good | 5 | \( 1 + (-2.13 + 0.693i)T + (4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (-1.40 + 4.33i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.242 - 0.746i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.320 + 0.232i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + (2.38 - 3.27i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-8.84 - 2.87i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.15 - 3.74i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-8.23 + 5.98i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 0.783iT - 43T^{2} \) |
| 47 | \( 1 + (1.65 + 2.28i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.77 - 8.54i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (3.80 - 5.24i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.57 - 4.84i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 2.26T + 67T^{2} \) |
| 71 | \( 1 + (0.360 + 1.10i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-10.9 - 7.94i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (4.85 + 1.57i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.78 + 8.55i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 18.2iT - 89T^{2} \) |
| 97 | \( 1 + (3.60 + 1.17i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.55526538467754816491754927485, −10.23181846709339783785671151445, −9.226092175919903382250324111185, −8.257874639944318563680866515114, −7.75838547699347829716382139556, −6.19961930854224855434830857453, −5.32578949819152640707698448075, −3.93671266031680483622072882811, −2.67554303701187940273954670151, −1.14945200477765924031548979180,
1.82184187300796732929207824251, 2.41950004165701242594454786101, 4.47050669280788973493416885643, 5.88818688441465687358662496865, 6.47074367044499405705949933660, 7.67728048896125312347783341548, 8.348127815516630632791616981144, 9.602839265828575708283013762905, 9.728522085363266685170001929031, 11.16646069749160755770254614550