L(s) = 1 | + (−1.28 − 0.581i)2-s + (−0.707 − 0.707i)3-s + (1.32 + 1.50i)4-s + (0.500 + 1.32i)6-s + (−1.41 + 1.41i)7-s + (−0.832 − 2.70i)8-s + 1.00i·9-s − 5.29i·11-s + (0.125 − 1.99i)12-s + (3.74 − 3.74i)13-s + (2.64 − 1.00i)14-s + (−0.5 + 3.96i)16-s + (0.581 − 1.28i)18-s − 5.29·19-s + 2.00·21-s + (−3.07 + 6.82i)22-s + ⋯ |
L(s) = 1 | + (−0.911 − 0.411i)2-s + (−0.408 − 0.408i)3-s + (0.661 + 0.750i)4-s + (0.204 + 0.540i)6-s + (−0.534 + 0.534i)7-s + (−0.294 − 0.955i)8-s + 0.333i·9-s − 1.59i·11-s + (0.0361 − 0.576i)12-s + (1.03 − 1.03i)13-s + (0.707 − 0.267i)14-s + (−0.125 + 0.992i)16-s + (0.137 − 0.303i)18-s − 1.21·19-s + 0.436·21-s + (−0.656 + 1.45i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.254621 - 0.492347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.254621 - 0.492347i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.28 + 0.581i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.41 - 1.41i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.29iT - 11T^{2} \) |
| 13 | \( 1 + (-3.74 + 3.74i)T - 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 + 5.29iT - 31T^{2} \) |
| 37 | \( 1 + (3.74 + 3.74i)T + 37iT^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (7.48 - 7.48i)T - 53iT^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + (-8.48 + 8.48i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-7.48 + 7.48i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + (7.48 + 7.48i)T + 97iT^{2} \) |
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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
−11.11366827078339260204414827973, −10.75000148874761125591235922589, −9.522510834844245996872367918424, −8.443259421179761906909354427913, −7.967479000495507401069749577781, −6.30348449728015799957283643771, −5.96712517968522044591196862250, −3.74548443100461685988092369663, −2.49344010133412599717401419337, −0.57158506166021333835700289376,
1.76988193963928689605102938317, 3.89748684666363939275992693397, 5.16944596390723789737226246894, 6.62532969115571404495917450050, 6.95400089430908119250936073257, 8.415726047008918984387926615853, 9.327173841893548444258978469204, 10.14885704861837157836428690361, 10.79977484790948380893976954149, 11.82757147536711738531306886771