L(s) = 1 | + (−0.618 − 1.90i)2-s + (5.32 + 3.86i)3-s + (−3.23 + 2.35i)4-s + (−1.54 + 4.75i)5-s + (4.06 − 12.5i)6-s + (7.43 − 5.40i)7-s + (6.47 + 4.70i)8-s + (5.04 + 15.5i)9-s + 10.0·10-s + (32.2 + 16.9i)11-s − 26.3·12-s + (24.6 + 75.7i)13-s + (−14.8 − 10.8i)14-s + (−26.6 + 19.3i)15-s + (4.94 − 15.2i)16-s + (−5.70 + 17.5i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (1.02 + 0.744i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (0.276 − 0.851i)6-s + (0.401 − 0.291i)7-s + (0.286 + 0.207i)8-s + (0.186 + 0.575i)9-s + 0.316·10-s + (0.885 + 0.465i)11-s − 0.633·12-s + (0.525 + 1.61i)13-s + (−0.284 − 0.206i)14-s + (−0.458 + 0.333i)15-s + (0.0772 − 0.237i)16-s + (−0.0813 + 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.90944 + 0.399816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.90944 + 0.399816i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.618 + 1.90i)T \) |
| 5 | \( 1 + (1.54 - 4.75i)T \) |
| 11 | \( 1 + (-32.2 - 16.9i)T \) |
good | 3 | \( 1 + (-5.32 - 3.86i)T + (8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + (-7.43 + 5.40i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-24.6 - 75.7i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (5.70 - 17.5i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-12.9 - 9.41i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 7.44T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-44.2 + 32.1i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (55.5 + 170. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-133. + 96.7i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (179. + 130. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (380. + 276. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (224. + 691. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (114. - 83.1i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (191. - 590. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 783.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (99.8 - 307. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-806. + 586. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-54.1 - 166. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-214. + 659. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 101.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-16.1 - 49.6i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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
−13.47881450648250839791448385984, −11.91374802976151544669028208453, −11.13323259048964494915362248490, −9.863337218339843916041457557804, −9.178836339133585218344912050477, −8.167067312005438211390957464408, −6.70706727599015435209827724511, −4.38498764097170279568610455069, −3.61806282192386613066974159009, −1.93252453882285229670585470730,
1.23185427506815647531564093111, 3.23650743115899426341393550059, 5.15462465326309848761268680779, 6.57012918759035079529729605687, 7.946636753421089880194442560691, 8.392619254499830148854726436639, 9.416407151085666820401739923210, 10.99196196346646940016724184708, 12.42705799078598520418494248061, 13.33156520281787727653218881462