| L(s) = 1 | + (−4.5 + 2.59i)3-s + (1.51 + 0.874i)5-s + (−2.69 − 48.9i)7-s + (13.5 − 23.3i)9-s + (63.6 + 110. i)11-s + 156. i·13-s − 9.09·15-s + (−245. + 142. i)17-s + (−378. − 218. i)19-s + (139. + 213. i)21-s + (249. − 431. i)23-s + (−310. − 538. i)25-s + 140. i·27-s − 1.43e3·29-s + (−633. + 365. i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.288i)3-s + (0.0606 + 0.0349i)5-s + (−0.0550 − 0.998i)7-s + (0.166 − 0.288i)9-s + (0.526 + 0.911i)11-s + 0.924i·13-s − 0.0404·15-s + (−0.851 + 0.491i)17-s + (−1.04 − 0.605i)19-s + (0.315 + 0.483i)21-s + (0.470 − 0.815i)23-s + (−0.497 − 0.861i)25-s + 0.192i·27-s − 1.70·29-s + (−0.659 + 0.380i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0715i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1522043314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1522043314\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 - 2.59i)T \) |
| 7 | \( 1 + (2.69 + 48.9i)T \) |
| good | 5 | \( 1 + (-1.51 - 0.874i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (-63.6 - 110. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 156. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (245. - 142. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (378. + 218. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-249. + 431. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + 1.43e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + (633. - 365. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (1.26e3 - 2.19e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 - 52.0iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 67.7T + 3.41e6T^{2} \) |
| 47 | \( 1 + (2.21e3 + 1.27e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-791. - 1.37e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (22.1 - 12.7i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.22e3 - 1.28e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (2.43e3 + 4.20e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 8.62e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (7.45e3 - 4.30e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (1.64e3 - 2.85e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 2.43e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (31.5 + 18.1i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 9.88e3iT - 8.85e7T^{2} \) |
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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
−12.58859209426841282727742206512, −11.44956388097677283944957662168, −10.65077834900315123919452356059, −9.720123188962218664021995476386, −8.640647047578301506015907452553, −7.05120070183709232062998386412, −6.47746746169095197431889073122, −4.71259244180323176023075778636, −3.97080414238868318179019961779, −1.82601848684543246667418138632,
0.05813843402144750011755659346, 1.92575908231232003563864467084, 3.57605346671671534470816514636, 5.37878517206672107852004699092, 6.02155403774548429151511319025, 7.36561656637314201583638982320, 8.611590870534924253830504562045, 9.466468276241047818230551093757, 10.95274204592531712055139134065, 11.49850729112816595900063486399
