| L(s) = 1 | + (3.25 + 8.93i)5-s + (0.410 + 2.32i)7-s + (−4.40 + 12.1i)11-s + (−12.2 − 10.2i)13-s + (−12.2 − 7.04i)17-s + (−3.29 − 5.70i)19-s + (25.9 + 4.58i)23-s + (−50.0 + 41.9i)25-s + (−0.977 − 1.16i)29-s + (−0.620 + 3.52i)31-s + (−19.4 + 11.2i)35-s + (−11.0 + 19.2i)37-s + (−31.4 + 37.5i)41-s + (78.8 + 28.6i)43-s + (34.9 − 6.16i)47-s + ⋯ |
| L(s) = 1 | + (0.650 + 1.78i)5-s + (0.0585 + 0.332i)7-s + (−0.400 + 1.10i)11-s + (−0.939 − 0.788i)13-s + (−0.717 − 0.414i)17-s + (−0.173 − 0.300i)19-s + (1.12 + 0.199i)23-s + (−2.00 + 1.67i)25-s + (−0.0337 − 0.0401i)29-s + (−0.0200 + 0.113i)31-s + (−0.555 + 0.320i)35-s + (−0.299 + 0.519i)37-s + (−0.767 + 0.914i)41-s + (1.83 + 0.667i)43-s + (0.743 − 0.131i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.693009 + 1.25429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.693009 + 1.25429i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-3.25 - 8.93i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-0.410 - 2.32i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (4.40 - 12.1i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (12.2 + 10.2i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (12.2 + 7.04i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (3.29 + 5.70i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-25.9 - 4.58i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (0.977 + 1.16i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (0.620 - 3.52i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (11.0 - 19.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (31.4 - 37.5i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-78.8 - 28.6i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-34.9 + 6.16i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 65.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-17.2 - 47.3i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (11.5 + 65.6i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-72.5 - 60.8i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (71.8 + 41.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (47.8 + 82.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (25.9 - 21.8i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (7.14 + 8.50i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-8.83 + 5.10i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-66.7 - 24.2i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.53053110929608427759533216889, −10.63741612479907311586616219692, −10.04365801919099571155168222832, −9.142349362673362010342066237069, −7.49797556312470926028453060263, −7.04682237194659878535863183090, −5.93405368964038470540316509527, −4.77109481382684271936073837455, −2.97766484435975823086271659191, −2.29241131469962979558437472831,
0.65334524719764672723802961076, 2.13520383478598351356189468749, 4.10044063230183553913677566657, 5.06664383942144658673112321369, 5.91212970063115661020994717295, 7.29095321170274164053118003849, 8.632703207811281118043427326916, 8.945005347023761154777995150448, 10.04011289179545041717910185771, 11.08550011941740702322622491478
