L(s) = 1 | + (−0.358 − 0.207i)2-s + (−0.914 − 1.58i)4-s + (1.46 − 2.53i)5-s + 1.58i·8-s + (−1.05 + 0.606i)10-s + (4.18 − 2.41i)11-s + 2.93i·13-s + (−1.49 + 2.59i)16-s + (−3.53 − 6.12i)17-s + (−5.07 − 2.93i)19-s − 5.35·20-s − 2·22-s + (1.73 + i)23-s + (−1.79 − 3.10i)25-s + (0.606 − 1.05i)26-s + ⋯ |
L(s) = 1 | + (−0.253 − 0.146i)2-s + (−0.457 − 0.791i)4-s + (0.655 − 1.13i)5-s + 0.560i·8-s + (−0.332 + 0.191i)10-s + (1.26 − 0.727i)11-s + 0.812i·13-s + (−0.374 + 0.649i)16-s + (−0.857 − 1.48i)17-s + (−1.16 − 0.672i)19-s − 1.19·20-s − 0.426·22-s + (0.361 + 0.208i)23-s + (−0.358 − 0.621i)25-s + (0.119 − 0.206i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.589381 - 0.972304i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.589381 - 0.972304i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.358 + 0.207i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.46 + 2.53i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.18 + 2.41i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 + (3.53 + 6.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.07 + 2.93i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 - i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.828iT - 29T^{2} \) |
| 31 | \( 1 + (5.07 - 2.93i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.70 + 4.68i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 + (-2.93 + 5.07i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.12 - 3.53i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.93 - 5.07i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.05 + 0.606i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.24 - 7.34i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.828iT - 71T^{2} \) |
| 73 | \( 1 + (-6.12 + 3.53i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.828 - 1.43i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + (5.60 - 9.71i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 7.07iT - 97T^{2} \) |
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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
−10.90525934838017594978547447975, −9.542130590926080234903550636847, −9.055159077324915474100747359452, −8.731853538563880585840083935235, −6.94025150909168473493777404697, −6.00044826500447758174708338709, −5.00498950744690053462465164552, −4.20175228854807131295539411652, −2.09119416216593469476841220880, −0.817435497941986455493466828646,
2.08536823957015810192403323028, 3.48877780656809221776394884904, 4.39230686741291378255483478267, 6.14408517421830221248132448322, 6.70189816540234651160349704120, 7.77278611897289496159593466822, 8.711169748258876767827426900381, 9.601825264177506138373908796044, 10.44812771320595618083206021564, 11.22879736680530777051515374524