| L(s) = 1 | + (−1.37 − 1.45i)2-s + (−0.219 + 3.99i)4-s + (3.09 − 1.12i)5-s + (−9.81 + 1.73i)7-s + (6.10 − 5.17i)8-s + (−5.89 − 2.94i)10-s + (0.511 − 1.40i)11-s + (7.50 + 6.29i)13-s + (16.0 + 11.8i)14-s + (−15.9 − 1.75i)16-s + (−2.21 + 3.83i)17-s + (−15.6 + 9.02i)19-s + (3.82 + 12.6i)20-s + (−2.74 + 1.18i)22-s + (−31.4 − 5.54i)23-s + ⋯ |
| L(s) = 1 | + (−0.687 − 0.726i)2-s + (−0.0547 + 0.998i)4-s + (0.619 − 0.225i)5-s + (−1.40 + 0.247i)7-s + (0.762 − 0.646i)8-s + (−0.589 − 0.294i)10-s + (0.0464 − 0.127i)11-s + (0.577 + 0.484i)13-s + (1.14 + 0.848i)14-s + (−0.993 − 0.109i)16-s + (−0.130 + 0.225i)17-s + (−0.822 + 0.474i)19-s + (0.191 + 0.630i)20-s + (−0.124 + 0.0540i)22-s + (−1.36 − 0.240i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0472 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0472 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.320639 + 0.336175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.320639 + 0.336175i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 + 1.45i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-3.09 + 1.12i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (9.81 - 1.73i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-0.511 + 1.40i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-7.50 - 6.29i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (2.21 - 3.83i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (15.6 - 9.02i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (31.4 + 5.54i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (17.2 - 14.4i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-42.5 - 7.50i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (35.6 - 61.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-2.00 - 1.68i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (15.2 - 41.8i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-8.16 + 1.43i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 54.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-31.7 - 87.3i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (14.4 + 81.7i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (45.3 - 53.9i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (95.6 + 55.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (28.1 + 48.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-68.8 - 82.0i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (21.4 + 25.5i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (9.14 + 15.8i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (69.4 + 25.2i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.67067578204064379603949813323, −10.38114567666972694923821934908, −9.891820565095949261401880393526, −8.998512103478919055861445146830, −8.224588796054812379233652636408, −6.74668575986538024489085697381, −5.99073230296121712319682106071, −4.17889523699734731704118190319, −3.05073156554528711753471466243, −1.68577767086935310354805551506,
0.26303294097430944233555020391, 2.27214210226752065869794983413, 3.98982438304290813907172768688, 5.69573280159241094252100978165, 6.31270462598368645306976607779, 7.18202865972096239561790034112, 8.366302320800697409289233780981, 9.351959744792775490878179066871, 10.07927252799433196032490205657, 10.66311245723300385060733864969
