L(s) = 1 | − 5.02·2-s + (4.35 − 2.83i)3-s + 17.2·4-s + 15.7i·5-s + (−21.8 + 14.2i)6-s − 18.9·7-s − 46.3·8-s + (10.8 − 24.7i)9-s − 79.3i·10-s − 36.5·11-s + (75.0 − 48.9i)12-s − 4.20i·13-s + 95.2·14-s + (44.8 + 68.7i)15-s + 95.1·16-s − 97.3i·17-s + ⋯ |
L(s) = 1 | − 1.77·2-s + (0.837 − 0.546i)3-s + 2.15·4-s + 1.41i·5-s + (−1.48 + 0.970i)6-s − 1.02·7-s − 2.05·8-s + (0.403 − 0.915i)9-s − 2.50i·10-s − 1.00·11-s + (1.80 − 1.17i)12-s − 0.0897i·13-s + 1.81·14-s + (0.771 + 1.18i)15-s + 1.48·16-s − 1.38i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.227411 - 0.365324i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.227411 - 0.365324i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.35 + 2.83i)T \) |
| 59 | \( 1 + (389. - 231. i)T \) |
good | 2 | \( 1 + 5.02T + 8T^{2} \) |
| 5 | \( 1 - 15.7iT - 125T^{2} \) |
| 7 | \( 1 + 18.9T + 343T^{2} \) |
| 11 | \( 1 + 36.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.20iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 97.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 44.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 190.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 131. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 206. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 106. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 214. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 213. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 444.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 624. iT - 1.48e5T^{2} \) |
| 61 | \( 1 + 437. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 648. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 809. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 39.3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 595.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 642.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 258.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.81e3iT - 9.12e5T^{2} \) |
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
−11.47423073398661350548271784136, −10.56279460866576672751510188273, −9.742252684023914629754873395228, −8.995136869555538527542511163654, −7.68540805174500131424552254438, −7.15378388783656422887728677912, −6.32680222433094734209950813543, −3.05339737355306789075037170358, −2.44560635793978728481055317178, −0.30774461172463386428779746569,
1.47063273166797278744096263859, 3.09317577375831927534316182230, 4.95796031686127627593432521999, 6.70509760253972808637778973650, 8.084974427577663006612473645737, 8.601445652302177716696051776928, 9.367046955644583029244122635996, 10.15351100539682428494246759491, 10.98174439966234650341714921492, 12.74022583779035667289077436839