L(s) = 1 | + (1.33 − 0.467i)2-s + (1.56 − 1.24i)4-s + (0.316 + 0.548i)5-s + (2.06 + 1.65i)7-s + (1.50 − 2.39i)8-s + (0.679 + 0.584i)10-s + (0.424 + 0.245i)11-s + 3.13i·13-s + (3.52 + 1.24i)14-s + (0.882 − 3.90i)16-s + (−0.987 − 0.569i)17-s + (−0.591 − 1.02i)19-s + (1.18 + 0.462i)20-s + (0.681 + 0.128i)22-s + (−2.80 − 4.85i)23-s + ⋯ |
L(s) = 1 | + (0.943 − 0.330i)2-s + (0.781 − 0.624i)4-s + (0.141 + 0.245i)5-s + (0.779 + 0.626i)7-s + (0.530 − 0.847i)8-s + (0.214 + 0.184i)10-s + (0.127 + 0.0738i)11-s + 0.868i·13-s + (0.942 + 0.332i)14-s + (0.220 − 0.975i)16-s + (−0.239 − 0.138i)17-s + (−0.135 − 0.234i)19-s + (0.263 + 0.103i)20-s + (0.145 + 0.0274i)22-s + (−0.584 − 1.01i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.69345 - 0.468101i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.69345 - 0.468101i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.33 + 0.467i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.06 - 1.65i)T \) |
good | 5 | \( 1 + (-0.316 - 0.548i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.424 - 0.245i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.13iT - 13T^{2} \) |
| 17 | \( 1 + (0.987 + 0.569i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.591 + 1.02i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.80 + 4.85i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.05T + 29T^{2} \) |
| 31 | \( 1 + (-5.40 - 3.11i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.53 - 3.77i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.06iT - 41T^{2} \) |
| 43 | \( 1 - 4.65T + 43T^{2} \) |
| 47 | \( 1 + (4.80 + 8.32i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.10 - 1.91i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (9.12 + 5.26i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.1 - 5.87i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.11 + 5.40i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 + (5.35 - 9.27i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.49 + 1.43i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14.1iT - 83T^{2} \) |
| 89 | \( 1 + (-9.12 + 5.26i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.03185145549341871480537832521, −10.28425888681648096375419577245, −9.175576684757458487984948492030, −8.162287682955007000885966574974, −6.87605636567788907258851465910, −6.18246632811643786779831822256, −4.99573104889599044943385228370, −4.27550459569983765702489186876, −2.80112041237427559703074530781, −1.76753972145455143152773777039,
1.70185367012443204608493761464, 3.29614341405226191123789458196, 4.33639503782377346234955887306, 5.29102589102078578071028887882, 6.13342647469686386199673653743, 7.42076868647757304889304224026, 7.897993348811470083800251215517, 9.045688333117190112511801167463, 10.39211410266237761969495268528, 11.08230656361373616870534447017