L(s) = 1 | + (−0.416 − 0.240i)5-s + (2.88 + 4.99i)11-s + 1.41i·13-s + (19.9 − 11.5i)17-s + (9.24 + 5.33i)19-s + (−3.29 + 5.71i)23-s + (−12.3 − 21.4i)25-s + 6.20·29-s + (−36.3 + 20.9i)31-s + (30.0 − 52.0i)37-s + 48.8i·41-s − 51.5·43-s + (16.6 + 9.62i)47-s + (41.0 + 71.1i)53-s − 2.77i·55-s + ⋯ |
L(s) = 1 | + (−0.0832 − 0.0480i)5-s + (0.261 + 0.453i)11-s + 0.109i·13-s + (1.17 − 0.678i)17-s + (0.486 + 0.280i)19-s + (−0.143 + 0.248i)23-s + (−0.495 − 0.858i)25-s + 0.213·29-s + (−1.17 + 0.676i)31-s + (0.812 − 1.40i)37-s + 1.19i·41-s − 1.19·43-s + (0.354 + 0.204i)47-s + (0.774 + 1.34i)53-s − 0.0503i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.019686155\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.019686155\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.416 + 0.240i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-2.88 - 4.99i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 1.41iT - 169T^{2} \) |
| 17 | \( 1 + (-19.9 + 11.5i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-9.24 - 5.33i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (3.29 - 5.71i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 6.20T + 841T^{2} \) |
| 31 | \( 1 + (36.3 - 20.9i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-30.0 + 52.0i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 48.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-16.6 - 9.62i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-41.0 - 71.1i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-80.1 + 46.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-4.32 - 2.49i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-1.10 - 1.91i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 80.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-12.0 + 6.95i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-32.4 + 56.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-90.2 - 52.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 31.7iT - 9.40e3T^{2} \) |
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
−9.318521049342390099469268015100, −8.244887122444758828170900313016, −7.58116176177180902294929797480, −6.84872863709508935940298292632, −5.83777786750128384503678920280, −5.11214519204189161213239083523, −4.09990545435017366520938564810, −3.23092761865294386225873804752, −2.06035506472599342981329837082, −0.839072205205657334634412721737,
0.75256839401589152096288609196, 1.96611724444540968159689209820, 3.28274256116156936556329216173, 3.88629839869279001422306877572, 5.16208271295150915711066259385, 5.77967754491328997321386561039, 6.71902922444387114892708633314, 7.58001241953730028109389269124, 8.263514110890544745913172457346, 9.073806970389173944560121469757