| L(s) = 1 | + (0.295 + 1.38i)2-s + (0.947 − 2.60i)3-s + (−1.82 + 0.818i)4-s + (1.56 − 0.275i)5-s + (3.88 + 0.540i)6-s + (−0.102 + 0.176i)7-s + (−1.67 − 2.28i)8-s + (−3.58 − 3.00i)9-s + (0.844 + 2.08i)10-s + (3.16 − 1.82i)11-s + (0.401 + 5.52i)12-s + (1.56 + 4.30i)13-s + (−0.274 − 0.0888i)14-s + (0.764 − 4.33i)15-s + (2.66 − 2.98i)16-s + (−3.79 + 3.18i)17-s + ⋯ |
| L(s) = 1 | + (0.209 + 0.977i)2-s + (0.547 − 1.50i)3-s + (−0.912 + 0.409i)4-s + (0.699 − 0.123i)5-s + (1.58 + 0.220i)6-s + (−0.0385 + 0.0668i)7-s + (−0.590 − 0.806i)8-s + (−1.19 − 1.00i)9-s + (0.266 + 0.658i)10-s + (0.953 − 0.550i)11-s + (0.115 + 1.59i)12-s + (0.434 + 1.19i)13-s + (−0.0734 − 0.0237i)14-s + (0.197 − 1.11i)15-s + (0.665 − 0.746i)16-s + (−0.919 + 0.771i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42398 + 0.00230100i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42398 + 0.00230100i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.295 - 1.38i)T \) |
| 19 | \( 1 + (3.94 + 1.85i)T \) |
| good | 3 | \( 1 + (-0.947 + 2.60i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-1.56 + 0.275i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.102 - 0.176i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.16 + 1.82i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.56 - 4.30i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (3.79 - 3.18i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.845 - 4.79i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (4.64 - 5.53i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.446 + 0.772i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.9iT - 37T^{2} \) |
| 41 | \( 1 + (-0.759 - 0.276i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.05 + 0.538i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.601 - 0.504i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-3.09 - 0.546i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (5.40 + 6.43i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (4.69 + 0.827i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-6.59 + 7.85i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.87 - 10.6i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.52 - 1.28i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (15.0 + 5.47i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.11 - 1.22i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 3.74i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 11.6i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.22354497700368294672411191606, −12.54030525749621746654160957430, −11.27043170010315239520833723962, −9.136299309930271009730082866153, −8.845168491830867136895502404265, −7.50632494257580481023562483959, −6.55869600959328822570977927645, −5.92523667709325685750157488116, −3.92290687128353384344937780635, −1.82149898547802887044437412707,
2.44739752799402204107986936560, 3.82232610769434532782970509136, 4.73023117266120267514705379509, 6.09402391455080297404032137964, 8.405031897499649360360515550945, 9.285008517720315740182927839114, 10.06398805036305356201490665741, 10.64639893880402063289728302162, 11.79345488897079151417756648834, 13.11723668654345057578289721131
