| L(s) = 1 | + (−0.394 + 0.0695i)2-s + (0.955 − 2.84i)3-s + (−3.60 + 1.31i)4-s + (4.12 + 4.91i)5-s + (−0.179 + 1.18i)6-s + (−6.75 − 11.7i)7-s + (2.72 − 1.57i)8-s + (−7.17 − 5.43i)9-s + (−1.97 − 1.65i)10-s − 9.70i·11-s + (0.286 + 11.5i)12-s + (3.92 + 3.29i)13-s + (3.48 + 4.15i)14-s + (17.9 − 7.03i)15-s + (10.7 − 9.06i)16-s + (−8.83 − 10.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.197 + 0.0347i)2-s + (0.318 − 0.947i)3-s + (−0.901 + 0.328i)4-s + (0.824 + 0.983i)5-s + (−0.0298 + 0.198i)6-s + (−0.965 − 1.67i)7-s + (0.340 − 0.196i)8-s + (−0.797 − 0.603i)9-s + (−0.197 − 0.165i)10-s − 0.882i·11-s + (0.0238 + 0.959i)12-s + (0.301 + 0.253i)13-s + (0.248 + 0.296i)14-s + (1.19 − 0.468i)15-s + (0.674 − 0.566i)16-s + (−0.519 − 0.619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.448505 - 0.812531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.448505 - 0.812531i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.955 + 2.84i)T \) |
| 19 | \( 1 + (18.9 - 1.92i)T \) |
| good | 2 | \( 1 + (0.394 - 0.0695i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-4.12 - 4.91i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (6.75 + 11.7i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 9.70iT - 121T^{2} \) |
| 13 | \( 1 + (-3.92 - 3.29i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (8.83 + 10.5i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (9.48 + 26.0i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-2.83 - 7.78i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 28.3T + 961T^{2} \) |
| 37 | \( 1 - 6.23T + 1.36e3T^{2} \) |
| 41 | \( 1 + (0.165 - 0.0290i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-43.2 - 15.7i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-14.5 - 39.9i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-91.8 - 16.1i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-39.5 + 108. i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (51.9 + 43.6i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (3.81 - 21.6i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (7.74 - 1.36i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-48.5 - 17.6i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-54.5 + 45.8i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (31.6 - 18.2i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (38.0 + 104. i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-26.3 - 149. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.61308709963220640030871944573, −10.95318872925002849067899990041, −10.16217090124687297267276023054, −9.116562542408436725298069318617, −7.985990496588928804653130904426, −6.83090481133494684195359888476, −6.26457924146767036788804402639, −4.06159014331105683791838968246, −2.84257033107337326401218673195, −0.58481901573861640393609837254,
2.23786223318249650717572086007, 4.13318682927559356590153157052, 5.33169604361440943714938426398, 5.95696887302132478230053909438, 8.461875082679319974257329791730, 8.982681625413963554216455081988, 9.625817705179570085609193080978, 10.35097687948270390253964556615, 12.06896933082302414575956774019, 13.01124044388849365935800499129
