| L(s) = 1 | + (−24.5 + 11.2i)3-s + 414. i·7-s + (475. − 552. i)9-s + 61.5i·11-s − 1.37e3i·13-s − 6.80e3·17-s + 7.82e3·19-s + (−4.66e3 − 1.01e4i)21-s − 7.45e3·23-s + (−5.47e3 + 1.89e4i)27-s − 424. i·29-s − 4.96e4·31-s + (−692. − 1.51e3i)33-s − 2.27e3i·37-s + (1.55e4 + 3.38e4i)39-s + ⋯ |
| L(s) = 1 | + (−0.909 + 0.416i)3-s + 1.20i·7-s + (0.652 − 0.757i)9-s + 0.0462i·11-s − 0.627i·13-s − 1.38·17-s + 1.14·19-s + (−0.503 − 1.09i)21-s − 0.613·23-s + (−0.277 + 0.960i)27-s − 0.0173i·29-s − 1.66·31-s + (−0.0192 − 0.0420i)33-s − 0.0448i·37-s + (0.261 + 0.570i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8547251062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8547251062\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (24.5 - 11.2i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 414. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 61.5iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.37e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 6.80e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 7.82e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 7.45e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 424. iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.96e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 2.27e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 5.62e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.31e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 8.47e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.66e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 4.00e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 8.36e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.25e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.46e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.71e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.66e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.01e6T + 3.26e11T^{2} \) |
| 89 | \( 1 + 9.55e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.19e6iT - 8.32e11T^{2} \) |
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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
−10.79581429698631605229397582948, −9.607385366615262097446262919581, −8.982530034949985954163154419482, −7.66826741351836444320279431303, −6.41805844119416664331306108604, −5.60172713377414064244881437980, −4.77014420361941640315752989828, −3.39343597167893629574038359101, −1.95102613103113085067479575019, −0.31557057615446091937318084787,
0.800885301069242763600943247503, 1.97534063386358202929437411312, 3.83559792894639300969937306781, 4.76978835094861129528850067308, 5.96519594297584659624216966358, 7.02719207340643440052525736884, 7.52143057428495640219888639798, 8.976690781018842044003132873312, 10.11880686624536867030291504078, 10.92474291425679013319214505519
