L(s) = 1 | − 6.04·5-s + 0.880·7-s − 12.2·11-s − 7.16i·13-s − 15.7·17-s + (18.2 + 5.29i)19-s − 19.1·23-s + 11.5·25-s + 51.7i·29-s + 32.5i·31-s − 5.32·35-s − 25.3i·37-s − 19.0i·41-s − 39.5·43-s − 24.4·47-s + ⋯ |
L(s) = 1 | − 1.20·5-s + 0.125·7-s − 1.11·11-s − 0.551i·13-s − 0.926·17-s + (0.960 + 0.278i)19-s − 0.832·23-s + 0.460·25-s + 1.78i·29-s + 1.04i·31-s − 0.152·35-s − 0.686i·37-s − 0.465i·41-s − 0.920·43-s − 0.520·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8232150857\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8232150857\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-18.2 - 5.29i)T \) |
good | 5 | \( 1 + 6.04T + 25T^{2} \) |
| 7 | \( 1 - 0.880T + 49T^{2} \) |
| 11 | \( 1 + 12.2T + 121T^{2} \) |
| 13 | \( 1 + 7.16iT - 169T^{2} \) |
| 17 | \( 1 + 15.7T + 289T^{2} \) |
| 23 | \( 1 + 19.1T + 529T^{2} \) |
| 29 | \( 1 - 51.7iT - 841T^{2} \) |
| 31 | \( 1 - 32.5iT - 961T^{2} \) |
| 37 | \( 1 + 25.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 19.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 39.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 24.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 13.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 65.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 22.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 18.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 76.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 85.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 150. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 29.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 127. iT - 9.40e3T^{2} \) |
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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
−8.401431496590892089339440556241, −7.88315875731517725565245076132, −7.29549998695995797363316124637, −6.43718703298028229258528943259, −5.25634407724876777220517134923, −4.83622704839155137586564224899, −3.65519190109230164622471173548, −3.12327608142168806085864723126, −1.85211860420611905141739433604, −0.39391122653925277636771213659,
0.48284087719310112163941601555, 2.05316911484560094800107288456, 2.99666653887109392572837996108, 4.04622314309375110841756498946, 4.57731135678785763599735685803, 5.53190930383481799447921661546, 6.49390295346706315869121217749, 7.31526180450427395378424197284, 8.061471798189314520848913776344, 8.314087841362571685204032072569