| L(s) = 1 | + (−5.32e4 + 2.55e4i)3-s − 9.13e6i·5-s + 9.41e7·7-s + (2.18e9 − 2.71e9i)9-s + 4.84e10i·11-s + 8.27e10·13-s + (2.33e11 + 4.86e11i)15-s − 2.05e12i·17-s − 4.83e12·19-s + (−5.01e12 + 2.40e12i)21-s − 3.14e13i·23-s + 1.18e13·25-s + (−4.67e13 + 2.00e14i)27-s − 2.13e14i·29-s − 1.22e15·31-s + ⋯ |
| L(s) = 1 | + (−0.901 + 0.432i)3-s − 0.935i·5-s + 0.333·7-s + (0.625 − 0.779i)9-s + 1.86i·11-s + 0.599·13-s + (0.404 + 0.843i)15-s − 1.02i·17-s − 0.787·19-s + (−0.300 + 0.144i)21-s − 0.758i·23-s + 0.124·25-s + (−0.227 + 0.973i)27-s − 0.507i·29-s − 1.49·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(0.3079696754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3079696754\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.32e4 - 2.55e4i)T \) |
| good | 5 | \( 1 + 9.13e6iT - 9.53e13T^{2} \) |
| 7 | \( 1 - 9.41e7T + 7.97e16T^{2} \) |
| 11 | \( 1 - 4.84e10iT - 6.72e20T^{2} \) |
| 13 | \( 1 - 8.27e10T + 1.90e22T^{2} \) |
| 17 | \( 1 + 2.05e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 + 4.83e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + 3.14e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 + 2.13e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 + 1.22e15T + 6.71e29T^{2} \) |
| 37 | \( 1 + 7.97e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + 5.40e15iT - 1.80e32T^{2} \) |
| 43 | \( 1 + 1.38e16T + 4.67e32T^{2} \) |
| 47 | \( 1 + 4.76e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 - 2.36e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 - 9.32e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 - 4.44e17T + 5.08e35T^{2} \) |
| 67 | \( 1 + 7.05e16T + 3.32e36T^{2} \) |
| 71 | \( 1 + 2.01e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 2.63e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + 1.54e19T + 8.96e37T^{2} \) |
| 83 | \( 1 + 2.96e19iT - 2.40e38T^{2} \) |
| 89 | \( 1 - 4.76e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + 2.44e18T + 5.43e39T^{2} \) |
| 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
−14.93912574154997524360144479793, −12.87536193365071035174909593477, −11.90413044722397658655444383623, −10.34606769169717031459247978501, −8.988605509971843153698989156317, −7.00582213687644774948311037774, −5.20431335444450435486063153561, −4.29314817398027493976154572343, −1.65598010785614388042635273233, −0.11042708615487611809666789746,
1.51993252389155556569403380168, 3.48159167986054862979675863812, 5.61709779243490293035588938755, 6.68092604008578795676552985502, 8.332626005414574003594677302655, 10.71074038607030131102357122204, 11.25924157564516036796331390664, 13.01934778125815722564912241902, 14.34907621873743000494101196253, 16.00636491208882759326927674986
