| L(s) = 1 | + (−1.07 − 0.918i)2-s + (1.08 − 2.98i)3-s + (0.313 + 1.97i)4-s + (−3.40 + 0.600i)5-s + (−3.91 + 2.21i)6-s + (1.17 − 2.02i)7-s + (1.47 − 2.41i)8-s + (−5.45 − 4.57i)9-s + (4.21 + 2.48i)10-s + (−1.02 + 0.592i)11-s + (6.24 + 1.21i)12-s + (0.845 + 2.32i)13-s + (−3.12 + 1.10i)14-s + (−1.91 + 10.8i)15-s + (−3.80 + 1.24i)16-s + (1.96 − 1.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.760 − 0.649i)2-s + (0.628 − 1.72i)3-s + (0.156 + 0.987i)4-s + (−1.52 + 0.268i)5-s + (−1.59 + 0.904i)6-s + (0.442 − 0.767i)7-s + (0.521 − 0.853i)8-s + (−1.81 − 1.52i)9-s + (1.33 + 0.785i)10-s + (−0.309 + 0.178i)11-s + (1.80 + 0.349i)12-s + (0.234 + 0.644i)13-s + (−0.835 + 0.295i)14-s + (−0.493 + 2.79i)15-s + (−0.950 + 0.310i)16-s + (0.476 − 0.400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0751581 - 0.694928i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0751581 - 0.694928i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.07 + 0.918i)T \) |
| 19 | \( 1 + (-2.52 + 3.55i)T \) |
| good | 3 | \( 1 + (-1.08 + 2.98i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (3.40 - 0.600i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.17 + 2.02i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.02 - 0.592i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.845 - 2.32i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.96 + 1.65i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.0208 + 0.118i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.07 + 3.65i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.669 + 1.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.90iT - 37T^{2} \) |
| 41 | \( 1 + (1.30 + 0.474i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-9.95 + 1.75i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (2.76 + 2.31i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-5.70 - 1.00i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.32 - 6.34i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (11.6 + 2.05i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.80 - 9.29i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0402 - 0.228i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.45 - 0.891i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-15.4 - 5.61i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.76 - 3.90i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.23 - 2.27i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (12.0 - 10.1i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.20999529785422041037023202301, −11.73082933830215074104629065767, −10.84175832857451788486860081627, −9.111421201424644423871450090551, −8.038939940754167364716049883847, −7.51880347132159206436095666257, −6.88622804455391962819813810525, −4.00096774837683427322842693188, −2.66929160708684008467215470037, −0.861632228717445327594340172858,
3.26637226657092439398139990013, 4.63014682787992624345826903145, 5.59480323625736458509507918426, 7.85003657628327841294619711552, 8.292287511753831071800347404753, 9.145015518972997987748308876148, 10.29623396938516697995443909540, 11.04732411483658301827074244808, 12.09019642204339296214213038089, 14.06762422714139998758561138847
