| L(s) = 1 | + (−1.92 + 0.528i)2-s + (0.0596 − 2.99i)3-s + (3.44 − 2.03i)4-s + (7.68 + 3.70i)5-s + (1.47 + 5.81i)6-s + (−7.69 − 6.13i)7-s + (−5.55 + 5.75i)8-s + (−8.99 − 0.358i)9-s + (−16.7 − 3.07i)10-s + (0.306 + 0.487i)11-s + (−5.91 − 10.4i)12-s + (−21.9 − 5.02i)13-s + (18.0 + 7.76i)14-s + (11.5 − 22.8i)15-s + (7.68 − 14.0i)16-s + (−8.45 − 8.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.964 + 0.264i)2-s + (0.0198 − 0.999i)3-s + (0.860 − 0.509i)4-s + (1.53 + 0.740i)5-s + (0.245 + 0.969i)6-s + (−1.09 − 0.876i)7-s + (−0.694 + 0.719i)8-s + (−0.999 − 0.0397i)9-s + (−1.67 − 0.307i)10-s + (0.0278 + 0.0442i)11-s + (−0.492 − 0.870i)12-s + (−1.69 − 0.386i)13-s + (1.29 + 0.554i)14-s + (0.771 − 1.52i)15-s + (0.480 − 0.877i)16-s + (−0.497 − 0.497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.338i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0886485 - 0.507767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0886485 - 0.507767i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.92 - 0.528i)T \) |
| 3 | \( 1 + (-0.0596 + 2.99i)T \) |
| 29 | \( 1 + (14.7 + 24.9i)T \) |
| good | 5 | \( 1 + (-7.68 - 3.70i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (7.69 + 6.13i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (-0.306 - 0.487i)T + (-52.4 + 109. i)T^{2} \) |
| 13 | \( 1 + (21.9 + 5.02i)T + (152. + 73.3i)T^{2} \) |
| 17 | \( 1 + (8.45 + 8.45i)T + 289iT^{2} \) |
| 19 | \( 1 + (0.492 - 4.37i)T + (-351. - 80.3i)T^{2} \) |
| 23 | \( 1 + (0.0694 - 0.0334i)T + (329. - 413. i)T^{2} \) |
| 31 | \( 1 + (8.87 + 25.3i)T + (-751. + 599. i)T^{2} \) |
| 37 | \( 1 + (-28.5 + 45.4i)T + (-593. - 1.23e3i)T^{2} \) |
| 41 | \( 1 + (18.2 - 18.2i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (34.1 + 11.9i)T + (1.44e3 + 1.15e3i)T^{2} \) |
| 47 | \( 1 + (36.0 - 22.6i)T + (958. - 1.99e3i)T^{2} \) |
| 53 | \( 1 + (-8.93 + 18.5i)T + (-1.75e3 - 2.19e3i)T^{2} \) |
| 59 | \( 1 - 40.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-7.01 - 62.3i)T + (-3.62e3 + 828. i)T^{2} \) |
| 67 | \( 1 + (24.4 + 107. i)T + (-4.04e3 + 1.94e3i)T^{2} \) |
| 71 | \( 1 + (-55.7 - 12.7i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (94.3 + 33.0i)T + (4.16e3 + 3.32e3i)T^{2} \) |
| 79 | \( 1 + (44.7 - 71.2i)T + (-2.70e3 - 5.62e3i)T^{2} \) |
| 83 | \( 1 + (-4.38 - 5.50i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (16.2 + 46.4i)T + (-6.19e3 + 4.93e3i)T^{2} \) |
| 97 | \( 1 + (16.1 - 142. i)T + (-9.17e3 - 2.09e3i)T^{2} \) |
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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
−10.53711138381410821359540501484, −9.791188515268616672637288604361, −9.328480389809475437850529811314, −7.72637290770993887417235956452, −7.02419024116469574651741536671, −6.40169549058766564529860406595, −5.51883105979834911046184342806, −2.86256022658718015047971259895, −2.05262281171843785074437118002, −0.28354270064560191460863961358,
2.06031895903128403283262840214, 3.03773001799746258954904757632, 4.89329523075405161199306183185, 5.87793481989027814439694768740, 6.81119039842418522469717504421, 8.626516808889895671656712754439, 9.097848570429652178098005461240, 9.903487817183537893817269609705, 10.11621362242284669286343223941, 11.53726784563863997517056456664
