| L(s) = 1 | + (−3.73 − 1.46i)2-s + (3.12 − 4.14i)3-s + (5.91 + 5.49i)4-s + (−2.72 − 1.85i)5-s + (−17.7 + 10.9i)6-s + (17.4 − 6.32i)7-s + (−0.126 − 0.263i)8-s + (−7.44 − 25.9i)9-s + (7.43 + 10.9i)10-s + (−6.32 − 41.9i)11-s + (41.2 − 7.38i)12-s + (20.3 − 16.1i)13-s + (−74.2 − 1.90i)14-s + (−16.2 + 5.49i)15-s + (−4.73 − 63.2i)16-s + (86.1 + 26.5i)17-s + ⋯ |
| L(s) = 1 | + (−1.31 − 0.517i)2-s + (0.601 − 0.798i)3-s + (0.739 + 0.686i)4-s + (−0.243 − 0.165i)5-s + (−1.20 + 0.742i)6-s + (0.939 − 0.341i)7-s + (−0.00560 − 0.0116i)8-s + (−0.275 − 0.961i)9-s + (0.235 + 0.344i)10-s + (−0.173 − 1.15i)11-s + (0.993 − 0.177i)12-s + (0.433 − 0.345i)13-s + (−1.41 − 0.0364i)14-s + (−0.278 + 0.0945i)15-s + (−0.0740 − 0.988i)16-s + (1.22 + 0.379i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.149902 - 0.903895i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149902 - 0.903895i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.12 + 4.14i)T \) |
| 7 | \( 1 + (-17.4 + 6.32i)T \) |
| good | 2 | \( 1 + (3.73 + 1.46i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (2.72 + 1.85i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (6.32 + 41.9i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-20.3 + 16.1i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-86.1 - 26.5i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (40.5 - 23.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-3.60 - 11.6i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (283. - 64.6i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (60.1 + 34.7i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-25.9 + 24.0i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (290. - 139. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-52.6 - 25.3i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-59.4 + 151. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-13.8 + 14.9i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-437. + 298. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-426. - 459. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-350. + 607. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-423. - 96.5i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (1.14e3 - 449. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (294. + 509. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-143. + 179. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-1.46e3 - 221. i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 + 1.15e3iT - 9.12e5T^{2} \) |
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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
−11.81421644574005684363622394365, −11.07541243246558094139464741864, −10.03026903621832636180134253636, −8.686279585538402349474906753005, −8.201691186078529560998226134370, −7.43619000149462803031913968332, −5.68754972665756691718919781293, −3.51484330406985758065231280619, −1.83475751683255423743761900417, −0.66644646605566230224299509560,
1.88886416324967347577900921689, 3.93713762225943376136973257259, 5.34814561440962407153074321787, 7.25560175486101273388836556890, 7.927284124625967433348489864853, 8.929392023880204701841184328194, 9.684985273437970120867174138920, 10.62799561284634891699684674737, 11.61074738362770638868093658691, 13.14773524256387587662871254972
