L(s) = 1 | + (1.22 − 1.22i)3-s + (5.24 − 5.24i)5-s + 5.32·7-s − 2.99i·9-s + (12.2 + 12.2i)11-s + (5.73 + 5.73i)13-s − 12.8i·15-s − 23.3·17-s + (11.7 − 11.7i)19-s + (6.52 − 6.52i)21-s − 5.80·23-s − 29.9i·25-s + (−3.67 − 3.67i)27-s + (−18.3 − 18.3i)29-s + 16.9i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (1.04 − 1.04i)5-s + 0.761·7-s − 0.333i·9-s + (1.11 + 1.11i)11-s + (0.441 + 0.441i)13-s − 0.856i·15-s − 1.37·17-s + (0.618 − 0.618i)19-s + (0.310 − 0.310i)21-s − 0.252·23-s − 1.19i·25-s + (−0.136 − 0.136i)27-s + (−0.634 − 0.634i)29-s + 0.545i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.37354 - 1.03456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.37354 - 1.03456i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
good | 5 | \( 1 + (-5.24 + 5.24i)T - 25iT^{2} \) |
| 7 | \( 1 - 5.32T + 49T^{2} \) |
| 11 | \( 1 + (-12.2 - 12.2i)T + 121iT^{2} \) |
| 13 | \( 1 + (-5.73 - 5.73i)T + 169iT^{2} \) |
| 17 | \( 1 + 23.3T + 289T^{2} \) |
| 19 | \( 1 + (-11.7 + 11.7i)T - 361iT^{2} \) |
| 23 | \( 1 + 5.80T + 529T^{2} \) |
| 29 | \( 1 + (18.3 + 18.3i)T + 841iT^{2} \) |
| 31 | \( 1 - 16.9iT - 961T^{2} \) |
| 37 | \( 1 + (15.3 - 15.3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 29.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-33.4 - 33.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 18.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-66.9 + 66.9i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (27.1 + 27.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (65.2 + 65.2i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (37.6 - 37.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 42.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 21.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-24.1 + 24.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 52.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 21.0T + 9.40e3T^{2} \) |
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
−11.15297521087253165821539422825, −9.718430581632679017347835445430, −9.152401541011923594766492580272, −8.470710388007541303750861868248, −7.18745626296951475454854459657, −6.28687988742998570077593464955, −5.02269940755024962817650319814, −4.17948966558699523179805271526, −2.12559985617973425805000920433, −1.36971579944281000441713733630,
1.67959182624567249717342122384, 2.97628138957115211081011557826, 4.08738266095117036798056012227, 5.61783675678265225288135876890, 6.33449001244123784161003803649, 7.49709363051240945994404931566, 8.715722820320391224368122151776, 9.317181143497099534917229784104, 10.53043440361281508645923877263, 10.94492043691830808314352929525