| L(s) = 1 | − 5.65i·2-s − 32.0·4-s + 1.84i·5-s − 12.6·7-s + 181. i·8-s + 10.4·10-s + 57.5i·11-s + 2.82e3·13-s + 71.7i·14-s + 1.02e3·16-s − 7.22e3i·17-s − 1.06e4·19-s − 59.0i·20-s + 325.·22-s + 1.33e4i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.0147i·5-s − 0.0369·7-s + 0.353i·8-s + 0.0104·10-s + 0.0432i·11-s + 1.28·13-s + 0.0261i·14-s + 0.250·16-s − 1.46i·17-s − 1.55·19-s − 0.00737i·20-s + 0.0305·22-s + 1.09i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8753498083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8753498083\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.65iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.84iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 12.6T + 1.17e5T^{2} \) |
| 11 | \( 1 - 57.5iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.82e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 7.22e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.06e4T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.33e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.55e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.73e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 6.04e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 8.71e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 7.58e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.14e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.47e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.15e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 4.02e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.81e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 3.01e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.12e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.43e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.51e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.02e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 4.95e5T + 8.32e11T^{2} \) |
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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
−11.23672122549611541842719483528, −10.41524502233195743845253141723, −9.245559551863369938317860861693, −8.419512737074587139157710915287, −7.01777686652666644988035859651, −5.69022903849481399012681470110, −4.35719618702046174811536819501, −3.15542975572988292767595512525, −1.73158108750643064100220864525, −0.25698189495752081993641004369,
1.50988004025498730216548283607, 3.47505651915012737597924076154, 4.67672492741733655036776501327, 6.09135309631908696951804896928, 6.77332939546294474406402121688, 8.399167664063316067652385274234, 8.698606550389151157310172324206, 10.33162172684918316869338605762, 11.01849227030185608431894722203, 12.66663906312822521136093805105
