| L(s) = 1 | + (0.717 + 1.21i)2-s + (−1.08 + 2.98i)3-s + (−0.970 + 1.74i)4-s + (3.40 − 0.600i)5-s + (−4.42 + 0.818i)6-s + (1.17 − 2.02i)7-s + (−2.82 + 0.0717i)8-s + (−5.45 − 4.57i)9-s + (3.17 + 3.72i)10-s + (1.02 − 0.592i)11-s + (−4.17 − 4.80i)12-s + (−0.845 − 2.32i)13-s + (3.31 − 0.0280i)14-s + (−1.91 + 10.8i)15-s + (−2.11 − 3.39i)16-s + (1.96 − 1.65i)17-s + ⋯ |
| L(s) = 1 | + (0.507 + 0.861i)2-s + (−0.628 + 1.72i)3-s + (−0.485 + 0.874i)4-s + (1.52 − 0.268i)5-s + (−1.80 + 0.334i)6-s + (0.442 − 0.767i)7-s + (−0.999 + 0.0253i)8-s + (−1.81 − 1.52i)9-s + (1.00 + 1.17i)10-s + (0.309 − 0.178i)11-s + (−1.20 − 1.38i)12-s + (−0.234 − 0.644i)13-s + (0.885 − 0.00748i)14-s + (−0.493 + 2.79i)15-s + (−0.528 − 0.848i)16-s + (0.476 − 0.400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.519761 + 1.26264i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.519761 + 1.26264i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.717 - 1.21i)T \) |
| 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
−13.74254324928631614093220874876, −12.52001256002444379023047089194, −11.17664704792845860328546280480, −10.08854304653782974276332380269, −9.522636855756292317906574280709, −8.313731819752573918957579622220, −6.46979321117945320623665142007, −5.48059217358989877544623321511, −4.83229349379335417041797759175, −3.51908843454143737163675252232,
1.67923192173290987390214577763, 2.39398764082280098202128572651, 5.18031254260211879136276956720, 6.01543584827256269666832250966, 6.80499615023528836096252622358, 8.570734395818371527202153744211, 9.731307329610796789152788254309, 11.01605530314907592443102563198, 11.80644535609681155901980123675, 12.65536618591934259406933948586
