L(s) = 1 | − 3.98i·2-s − 8.48·3-s − 7.85·4-s − 2.25i·5-s + 33.7i·6-s + 28.5i·7-s − 0.583i·8-s + 45.0·9-s − 8.99·10-s − 11i·11-s + 66.6·12-s + (−30.3 + 35.6i)13-s + 113.·14-s + 19.1i·15-s − 65.1·16-s + 120.·17-s + ⋯ |
L(s) = 1 | − 1.40i·2-s − 1.63·3-s − 0.981·4-s − 0.201i·5-s + 2.29i·6-s + 1.54i·7-s − 0.0257i·8-s + 1.66·9-s − 0.284·10-s − 0.301i·11-s + 1.60·12-s + (−0.648 + 0.761i)13-s + 2.17·14-s + 0.329i·15-s − 1.01·16-s + 1.72·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.760332 - 0.279741i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.760332 - 0.279741i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + 11iT \) |
| 13 | \( 1 + (30.3 - 35.6i)T \) |
good | 2 | \( 1 + 3.98iT - 8T^{2} \) |
| 3 | \( 1 + 8.48T + 27T^{2} \) |
| 5 | \( 1 + 2.25iT - 125T^{2} \) |
| 7 | \( 1 - 28.5iT - 343T^{2} \) |
| 17 | \( 1 - 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 92.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 314. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 127. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 31.1iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 447.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 188. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 485.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 201. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 15.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 170. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 7.43iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 779. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 888.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 583. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 297. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 836. iT - 9.12e5T^{2} \) |
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
−12.11251836739927543703647387033, −11.80236895746107592886897133073, −10.77066186651674829285737445776, −9.867067080890036972499636792636, −8.820791621169353648383174271412, −6.81706943123315001323258183662, −5.60502128877701152225708300979, −4.68129456214885778163235568193, −2.76211554161946493589882531017, −1.08474531291602469502790453967,
0.62866048271066784095279536431, 4.20807675645982664400449822345, 5.40037486251271498841454512718, 6.17661468854812332075763186250, 7.32032461110767339592047705505, 7.78863512981266011788746591068, 10.02756463546067254916443746161, 10.51668602099671395933432834803, 11.78543494184240111852257766741, 12.73424801377646042931064072898