L(s) = 1 | + (−0.959 + 0.281i)3-s + (−2.20 − 1.41i)5-s + (−0.514 − 3.58i)7-s + (0.841 − 0.540i)9-s + (−2.22 + 4.87i)11-s + (−0.416 + 2.89i)13-s + (2.51 + 0.737i)15-s + (5.05 + 5.83i)17-s + (−0.423 + 0.488i)19-s + (1.50 + 3.29i)21-s + (−0.961 + 4.69i)23-s + (0.769 + 1.68i)25-s + (−0.654 + 0.755i)27-s + (−2.40 − 2.77i)29-s + (−5.05 − 1.48i)31-s + ⋯ |
L(s) = 1 | + (−0.553 + 0.162i)3-s + (−0.984 − 0.632i)5-s + (−0.194 − 1.35i)7-s + (0.280 − 0.180i)9-s + (−0.671 + 1.47i)11-s + (−0.115 + 0.802i)13-s + (0.648 + 0.190i)15-s + (1.22 + 1.41i)17-s + (−0.0971 + 0.112i)19-s + (0.328 + 0.718i)21-s + (−0.200 + 0.979i)23-s + (0.153 + 0.337i)25-s + (−0.126 + 0.145i)27-s + (−0.446 − 0.515i)29-s + (−0.907 − 0.266i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.254213 + 0.361233i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.254213 + 0.361233i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (0.961 - 4.69i)T \) |
good | 5 | \( 1 + (2.20 + 1.41i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.514 + 3.58i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (2.22 - 4.87i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.416 - 2.89i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-5.05 - 5.83i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (0.423 - 0.488i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (2.40 + 2.77i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (5.05 + 1.48i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (2.76 - 1.77i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-10.4 - 6.70i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (6.82 - 2.00i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + (0.257 + 1.78i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.444 + 3.08i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-5.55 - 1.63i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-5.49 - 12.0i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.62 - 3.56i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-4.29 + 4.95i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.44 + 10.0i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (2.72 - 1.75i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (12.3 - 3.63i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (7.03 + 4.52i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.09130464904378982506101897931, −10.09165553347851180044741974866, −9.646960944779212672075527403310, −8.052283881204790688612507102346, −7.59083717898238815547239138675, −6.66756025035883733167583638468, −5.30953702879633456064944952028, −4.27637770997671732827148959443, −3.78865014515493978512310547548, −1.49692910152501995811444474938,
0.28125984242662832349062600526, 2.72053274014652747138224914732, 3.45757263271436020317727885845, 5.26689563052849501428411564252, 5.68895683693835005522515216648, 6.92366793241536087852629172531, 7.84994422193644617046426226314, 8.576663394662761006921647972390, 9.698117765145480448881491524691, 10.88351051736937395436966162802