| L(s) = 1 | + (−6.13 − 2.95i)2-s + (1.76 − 2.20i)3-s + (8.94 + 11.2i)4-s + (−38.8 − 18.7i)5-s + (−17.3 + 8.34i)6-s + (0.981 − 1.23i)7-s + (26.7 + 117. i)8-s + (52.2 + 229. i)9-s + (183. + 229. i)10-s + (−96.3 + 422. i)11-s + 40.5·12-s + (−44.0 + 192. i)13-s + (−9.65 + 4.64i)14-s + (−109. + 52.8i)15-s + (284. − 1.24e3i)16-s − 978.·17-s + ⋯ |
| L(s) = 1 | + (−1.08 − 0.522i)2-s + (0.113 − 0.141i)3-s + (0.279 + 0.350i)4-s + (−0.695 − 0.334i)5-s + (−0.196 + 0.0946i)6-s + (0.00756 − 0.00948i)7-s + (0.147 + 0.647i)8-s + (0.215 + 0.942i)9-s + (0.578 + 0.725i)10-s + (−0.240 + 1.05i)11-s + 0.0812·12-s + (−0.0722 + 0.316i)13-s + (−0.0131 + 0.00633i)14-s + (−0.125 + 0.0606i)15-s + (0.277 − 1.21i)16-s − 0.821·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0191 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0191 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.223686 + 0.228010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.223686 + 0.228010i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (2.27e3 - 3.91e3i)T \) |
| good | 2 | \( 1 + (6.13 + 2.95i)T + (19.9 + 25.0i)T^{2} \) |
| 3 | \( 1 + (-1.76 + 2.20i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (38.8 + 18.7i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 + (-0.981 + 1.23i)T + (-3.73e3 - 1.63e4i)T^{2} \) |
| 11 | \( 1 + (96.3 - 422. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (44.0 - 192. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + 978.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (85.9 + 107. i)T + (-5.50e5 + 2.41e6i)T^{2} \) |
| 23 | \( 1 + (1.79e3 - 863. i)T + (4.01e6 - 5.03e6i)T^{2} \) |
| 31 | \( 1 + (-2.20e3 - 1.05e3i)T + (1.78e7 + 2.23e7i)T^{2} \) |
| 37 | \( 1 + (390. + 1.70e3i)T + (-6.24e7 + 3.00e7i)T^{2} \) |
| 41 | \( 1 + 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (5.64e3 - 2.71e3i)T + (9.16e7 - 1.14e8i)T^{2} \) |
| 47 | \( 1 + (-1.21e3 + 5.34e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-1.28e4 - 6.21e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + 4.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-1.60e4 + 2.01e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (3.31e3 + 1.45e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (8.45e3 - 3.70e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-4.63e4 + 2.23e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 + (-1.28e4 - 5.62e4i)T + (-2.77e9 + 1.33e9i)T^{2} \) |
| 83 | \( 1 + (2.35e4 + 2.95e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-8.78e4 - 4.23e4i)T + (3.48e9 + 4.36e9i)T^{2} \) |
| 97 | \( 1 + (1.72e4 + 2.16e4i)T + (-1.91e9 + 8.37e9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.62742198280976195578339979267, −15.42751802139965641375233998441, −13.84067973690423467191478660411, −12.32962617169701261702405474687, −11.06568044220388333797156387473, −9.895274311795311852223190233553, −8.517390656264591196291395016497, −7.42577679905088282177537080912, −4.73215842289693608540487769174, −1.94455202723815112691425954036,
0.28258515357664512750494266377, 3.72211750134074184597878759209, 6.43837872144522926150696320977, 7.85047093266875918688405877798, 8.928430139757152448919883781179, 10.28006085555777230413346394163, 11.73332798598262178389738379696, 13.34348636138528997812458682337, 15.11461139896773407563495919133, 15.82733952752665875570444738402
