L(s) = 1 | + (−0.0912 − 1.41i)2-s + (−0.707 + 0.707i)3-s + (−1.98 + 0.257i)4-s + (1.06 + 0.933i)6-s + (1.86 + 1.86i)7-s + (0.544 + 2.77i)8-s − 1.00i·9-s + 0.728i·11-s + (1.22 − 1.58i)12-s + (3.12 + 3.12i)13-s + (2.46 − 2.80i)14-s + (3.86 − 1.02i)16-s + (−1.12 + 1.12i)17-s + (−1.41 + 0.0912i)18-s + 3.73·19-s + ⋯ |
L(s) = 1 | + (−0.0645 − 0.997i)2-s + (−0.408 + 0.408i)3-s + (−0.991 + 0.128i)4-s + (0.433 + 0.381i)6-s + (0.705 + 0.705i)7-s + (0.192 + 0.981i)8-s − 0.333i·9-s + 0.219i·11-s + (0.352 − 0.457i)12-s + (0.866 + 0.866i)13-s + (0.658 − 0.749i)14-s + (0.966 − 0.255i)16-s + (−0.272 + 0.272i)17-s + (−0.332 + 0.0215i)18-s + 0.856·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07343 - 0.0551590i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07343 - 0.0551590i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0912 + 1.41i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.86 - 1.86i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.728iT - 11T^{2} \) |
| 13 | \( 1 + (-3.12 - 3.12i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.12 - 1.12i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 + (-5.83 + 5.83i)T - 23iT^{2} \) |
| 29 | \( 1 - 2.64iT - 29T^{2} \) |
| 31 | \( 1 - 6.01iT - 31T^{2} \) |
| 37 | \( 1 + (3.12 - 3.12i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + (5.10 - 5.10i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.09 + 2.09i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.484 + 0.484i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + (-5.10 - 5.10i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.1iT - 71T^{2} \) |
| 73 | \( 1 + (3.96 + 3.96i)T + 73iT^{2} \) |
| 79 | \( 1 + 7.11T + 79T^{2} \) |
| 83 | \( 1 + (3.55 - 3.55i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.03iT - 89T^{2} \) |
| 97 | \( 1 + (-12.5 + 12.5i)T - 97iT^{2} \) |
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
−11.57997187946327920731164566975, −10.97408068177144080020065719727, −10.04494354314135184464727488079, −8.929808650818418563206185918589, −8.443296103579137786047732580883, −6.74697412839849587978367286484, −5.31051580732662035429632687205, −4.55493245404450818296464370575, −3.19485994822676857198822705381, −1.58987505401084893253416031023,
1.02625442251022805317732936383, 3.63456712892434686927069440608, 5.01559884467126158865370665600, 5.80190763743849449574041827384, 7.02802658580318381656248830191, 7.71477950692649182000023515189, 8.603540112636733830065780931293, 9.787104444262962092525543273448, 10.86004132267269294669785148372, 11.66320359684530215653439218144