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
−11.01815306000341755132345924124, −10.25109866015726731249827727510, −9.107389173462187022135486717426, −8.355653151014075936265107631590, −7.10445683547654164913608012537, −6.20968903884219485760496342323, −5.32778059396273007471710717484, −3.90937386246887672935305079755, −2.31549592201734614164095251155, −1.23089075047884273699070865124,
1.20443690188441578148274457526, 3.61874737023549836158527001077, 4.62707400023782998095571906663, 5.73339831476850229193672694271, 6.28877864838528993793563037247, 7.42871749806226011807148972992, 8.710685078051641320214420291270, 9.494641616430294163636401725212, 9.999907183403626716433684876869, 11.29981781332715567954890933441