| L(s) = 1 | + (0.841 − 2.03i)2-s + (0.134 + 0.0897i)3-s + (−0.590 − 0.590i)4-s + (5.26 − 1.04i)5-s + (0.295 − 0.197i)6-s + (−6.12 − 1.21i)7-s + (6.42 − 2.66i)8-s + (−3.43 − 8.29i)9-s + (2.30 − 11.5i)10-s + (8.11 + 12.1i)11-s + (−0.0263 − 0.132i)12-s + (4.79 − 4.79i)13-s + (−7.62 + 11.4i)14-s + (0.801 + 0.331i)15-s − 18.6i·16-s + ⋯ |
| L(s) = 1 | + (0.420 − 1.01i)2-s + (0.0447 + 0.0299i)3-s + (−0.147 − 0.147i)4-s + (1.05 − 0.209i)5-s + (0.0492 − 0.0329i)6-s + (−0.874 − 0.174i)7-s + (0.803 − 0.332i)8-s + (−0.381 − 0.921i)9-s + (0.230 − 1.15i)10-s + (0.737 + 1.10i)11-s + (−0.00219 − 0.0110i)12-s + (0.369 − 0.369i)13-s + (−0.544 + 0.815i)14-s + (0.0534 + 0.0221i)15-s − 1.16i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0637 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0637 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.70761 - 1.82010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70761 - 1.82010i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (-0.841 + 2.03i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-0.134 - 0.0897i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (-5.26 + 1.04i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (6.12 + 1.21i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-8.11 - 12.1i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-4.79 + 4.79i)T - 169iT^{2} \) |
| 19 | \( 1 + (-9.56 + 23.0i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-10.8 + 7.27i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-6.44 - 32.3i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-0.674 + 1.00i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (38.6 + 25.8i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (30.8 + 6.13i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-27.8 - 67.3i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (10.4 - 10.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-1.97 + 4.77i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (26.1 - 10.8i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (16.1 - 81.0i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 44.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-48.1 - 32.1i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-1.32 + 0.262i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-16.5 - 24.7i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-62.2 - 25.7i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-90.1 - 90.1i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (13.1 + 66.3i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−11.46282639427798933390526112435, −10.40576760591266303785693225421, −9.588095857211295029905627052690, −9.019724762083805645890125073079, −7.14265873020913941628362104904, −6.39534186633329936446977345399, −5.00772181172238937866767041398, −3.68779911640879025529139467490, −2.68953825417504450808717400725, −1.23683936116523171474269761824,
1.86192050031070993800125255397, 3.48656985645565224278895354524, 5.19254471875338142332847437838, 6.04790258029676149605866608955, 6.49546208972251602098454960037, 7.80912498836955491070306941114, 8.835472066217511735397646872198, 9.934874133384483757342920872779, 10.77597133175373155721617290270, 11.84658625857913315391936337539
