L(s) = 1 | + (−1.12 + 0.854i)2-s + (0.540 − 1.92i)4-s + (0.763 + 0.763i)5-s + 1.33·7-s + (1.03 + 2.63i)8-s + (−1.51 − 0.208i)10-s + (1.95 − 1.95i)11-s + (4.18 + 4.18i)13-s + (−1.50 + 1.14i)14-s + (−3.41 − 2.08i)16-s + 4.03i·17-s + (−4.26 + 4.26i)19-s + (1.88 − 1.05i)20-s + (−0.534 + 3.88i)22-s − 8.86i·23-s + ⋯ |
L(s) = 1 | + (−0.796 + 0.604i)2-s + (0.270 − 0.962i)4-s + (0.341 + 0.341i)5-s + 0.505·7-s + (0.366 + 0.930i)8-s + (−0.478 − 0.0658i)10-s + (0.590 − 0.590i)11-s + (1.16 + 1.16i)13-s + (−0.402 + 0.305i)14-s + (−0.854 − 0.520i)16-s + 0.978i·17-s + (−0.979 + 0.979i)19-s + (0.421 − 0.236i)20-s + (−0.113 + 0.827i)22-s − 1.84i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.804467 + 0.340636i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.804467 + 0.340636i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.12 - 0.854i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.763 - 0.763i)T + 5iT^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 + (-1.95 + 1.95i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.18 - 4.18i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.03iT - 17T^{2} \) |
| 19 | \( 1 + (4.26 - 4.26i)T - 19iT^{2} \) |
| 23 | \( 1 + 8.86iT - 23T^{2} \) |
| 29 | \( 1 + (-1.23 + 1.23i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.87iT - 31T^{2} \) |
| 37 | \( 1 + (-0.434 + 0.434i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.81T + 41T^{2} \) |
| 43 | \( 1 + (5.49 + 5.49i)T + 43iT^{2} \) |
| 47 | \( 1 + 3.20T + 47T^{2} \) |
| 53 | \( 1 + (-4.06 - 4.06i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.71 - 4.71i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.26 - 3.26i)T + 61iT^{2} \) |
| 67 | \( 1 + (5.44 - 5.44i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.76iT - 71T^{2} \) |
| 73 | \( 1 + 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 11.1iT - 79T^{2} \) |
| 83 | \( 1 + (-9.73 - 9.73i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.64T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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
−13.59524550415990105324905923981, −11.97221996076481920964552251051, −10.88648653664541068857094963459, −10.21144333281829946770051724021, −8.728541671341192684329105713785, −8.313254955761947082258923069756, −6.56795340910996606679720161616, −6.11077821056900959465925483579, −4.25234419758226984170054508496, −1.80752032818315695640285654960,
1.50493045485174487212830001669, 3.36733806326437601363928003519, 5.03754162128245509075340982690, 6.74876185229146804235629505709, 7.968055480975901492152817385913, 8.951383488323182150100206686155, 9.817660570765216787105988975598, 11.00200318448600736634277141639, 11.68390651071278090029548774368, 12.94446861942873579768804166143