| L(s) = 1 | − 1.74·2-s + (3.35 + 15.2i)3-s − 28.9·4-s + 25i·5-s + (−5.85 − 26.5i)6-s + 181. i·7-s + 106.·8-s + (−220. + 102. i)9-s − 43.6i·10-s + (−308. + 256. i)11-s + (−97.1 − 440. i)12-s + 507. i·13-s − 316. i·14-s + (−380. + 83.8i)15-s + 741.·16-s + 741.·17-s + ⋯ |
| L(s) = 1 | − 0.308·2-s + (0.215 + 0.976i)3-s − 0.904·4-s + 0.447i·5-s + (−0.0663 − 0.301i)6-s + 1.39i·7-s + 0.587·8-s + (−0.907 + 0.420i)9-s − 0.137i·10-s + (−0.768 + 0.639i)11-s + (−0.194 − 0.883i)12-s + 0.832i·13-s − 0.431i·14-s + (−0.436 + 0.0962i)15-s + 0.723·16-s + 0.621·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.790i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8322533295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8322533295\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.35 - 15.2i)T \) |
| 5 | \( 1 - 25iT \) |
| 11 | \( 1 + (308. - 256. i)T \) |
| good | 2 | \( 1 + 1.74T + 32T^{2} \) |
| 7 | \( 1 - 181. iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 507. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 741.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.72e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 418. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.05e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.88e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.66e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.00e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 8.48e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.09e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.11e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 7.44e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 9.35e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 4.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.00e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.84e4T + 8.58e9T^{2} \) |
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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
−12.48385802681673283863660376478, −11.47387791152459935207025967626, −10.16341389004237996776973483764, −9.617474498024596115371667265482, −8.682156047136402655536955930665, −7.79043558144613049205850037212, −5.84487652084884543642817652164, −4.95099063084083006557759668290, −3.68785124582779035292487384443, −2.23649566281595442450203160876,
0.37358034589639749477358555097, 1.02632398186815662238566388955, 3.12887529098452478308080391682, 4.59401665461246944190371749780, 5.89091957306956371672181454789, 7.51531528397541956084792242195, 7.962994480069735214822832635835, 9.068234651950124629164340715165, 10.21198342149445745961165541361, 11.19490998186762201210965798028
