L(s) = 1 | + (−0.309 + 0.951i)2-s + (−1.15 + 1.29i)3-s + (−0.809 − 0.587i)4-s + (1.45 − 0.471i)5-s + (−0.875 − 1.49i)6-s + (0.587 − 0.809i)7-s + (0.809 − 0.587i)8-s + (−0.349 − 2.97i)9-s + 1.52i·10-s + (3.21 − 0.816i)11-s + (1.69 − 0.370i)12-s + (3.83 + 1.24i)13-s + (0.587 + 0.809i)14-s + (−1.05 + 2.41i)15-s + (0.309 + 0.951i)16-s + (−1.92 − 5.93i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.664 + 0.747i)3-s + (−0.404 − 0.293i)4-s + (0.648 − 0.210i)5-s + (−0.357 − 0.610i)6-s + (0.222 − 0.305i)7-s + (0.286 − 0.207i)8-s + (−0.116 − 0.993i)9-s + 0.482i·10-s + (0.969 − 0.246i)11-s + (0.488 − 0.106i)12-s + (1.06 + 0.345i)13-s + (0.157 + 0.216i)14-s + (−0.273 + 0.624i)15-s + (0.0772 + 0.237i)16-s + (−0.467 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03197 + 0.611919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03197 + 0.611919i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (1.15 - 1.29i)T \) |
| 7 | \( 1 + (-0.587 + 0.809i)T \) |
| 11 | \( 1 + (-3.21 + 0.816i)T \) |
good | 5 | \( 1 + (-1.45 + 0.471i)T + (4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (-3.83 - 1.24i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.92 + 5.93i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-4.38 - 6.03i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 0.184iT - 23T^{2} \) |
| 29 | \( 1 + (2.12 + 1.54i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.712 + 2.19i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.41 - 2.48i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (2.03 - 1.47i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.14iT - 43T^{2} \) |
| 47 | \( 1 + (-6.08 - 8.37i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.46 + 0.477i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.34 + 4.60i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-6.79 + 2.20i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 0.875T + 67T^{2} \) |
| 71 | \( 1 + (-5.30 + 1.72i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.87 - 6.70i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (9.19 + 2.98i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.95 + 6.02i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 17.2iT - 89T^{2} \) |
| 97 | \( 1 + (-4.34 + 13.3i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.29981781332715567954890933441, −9.999907183403626716433684876869, −9.494641616430294163636401725212, −8.710685078051641320214420291270, −7.42871749806226011807148972992, −6.28877864838528993793563037247, −5.73339831476850229193672694271, −4.62707400023782998095571906663, −3.61874737023549836158527001077, −1.20443690188441578148274457526,
1.23089075047884273699070865124, 2.31549592201734614164095251155, 3.90937386246887672935305079755, 5.32778059396273007471710717484, 6.20968903884219485760496342323, 7.10445683547654164913608012537, 8.355653151014075936265107631590, 9.107389173462187022135486717426, 10.25109866015726731249827727510, 11.01815306000341755132345924124