| L(s) = 1 | + (0.383 + 0.278i)2-s + (1.11 + 2.35i)3-s + (−0.548 − 1.68i)4-s + (−1.78 + 2.16i)5-s + (−0.231 + 1.21i)6-s + (−0.184 + 2.93i)7-s + (0.553 − 1.70i)8-s + (−2.42 + 2.92i)9-s + (−1.28 + 0.330i)10-s + (1.92 − 4.08i)11-s + (3.37 − 3.16i)12-s + (4.85 − 0.613i)13-s + (−0.889 + 1.07i)14-s + (−7.08 − 1.81i)15-s + (−2.18 + 1.58i)16-s + (−0.126 − 2.01i)17-s + ⋯ |
| L(s) = 1 | + (0.271 + 0.197i)2-s + (0.641 + 1.36i)3-s + (−0.274 − 0.844i)4-s + (−0.799 + 0.966i)5-s + (−0.0946 + 0.496i)6-s + (−0.0697 + 1.10i)7-s + (0.195 − 0.602i)8-s + (−0.807 + 0.976i)9-s + (−0.407 + 0.104i)10-s + (0.580 − 1.23i)11-s + (0.974 − 0.914i)12-s + (1.34 − 0.170i)13-s + (−0.237 + 0.287i)14-s + (−1.82 − 0.469i)15-s + (−0.546 + 0.396i)16-s + (−0.0307 − 0.488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0942 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0942 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.982349 + 0.893778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.982349 + 0.893778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-10.7 - 5.87i)T \) |
| good | 2 | \( 1 + (-0.383 - 0.278i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.11 - 2.35i)T + (-1.91 + 2.31i)T^{2} \) |
| 5 | \( 1 + (1.78 - 2.16i)T + (-0.936 - 4.91i)T^{2} \) |
| 7 | \( 1 + (0.184 - 2.93i)T + (-6.94 - 0.877i)T^{2} \) |
| 11 | \( 1 + (-1.92 + 4.08i)T + (-7.01 - 8.47i)T^{2} \) |
| 13 | \( 1 + (-4.85 + 0.613i)T + (12.5 - 3.23i)T^{2} \) |
| 17 | \( 1 + (0.126 + 2.01i)T + (-16.8 + 2.13i)T^{2} \) |
| 19 | \( 1 + (-0.727 - 2.23i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.02 + 6.22i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.49 + 3.56i)T + (15.5 - 24.4i)T^{2} \) |
| 31 | \( 1 + (4.51 - 7.12i)T + (-13.1 - 28.0i)T^{2} \) |
| 37 | \( 1 + (6.70 - 0.846i)T + (35.8 - 9.20i)T^{2} \) |
| 41 | \( 1 + (-1.07 + 5.64i)T + (-38.1 - 15.0i)T^{2} \) |
| 43 | \( 1 + (0.584 + 9.28i)T + (-42.6 + 5.38i)T^{2} \) |
| 47 | \( 1 + (0.412 - 2.16i)T + (-43.6 - 17.3i)T^{2} \) |
| 53 | \( 1 + (2.15 + 2.02i)T + (3.32 + 52.8i)T^{2} \) |
| 59 | \( 1 + (8.84 - 6.42i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.77 - 3.77i)T + (-38.8 - 47.0i)T^{2} \) |
| 67 | \( 1 + (-4.54 - 5.49i)T + (-12.5 + 65.8i)T^{2} \) |
| 71 | \( 1 + (0.292 - 4.64i)T + (-70.4 - 8.89i)T^{2} \) |
| 73 | \( 1 + (-0.662 + 10.5i)T + (-72.4 - 9.14i)T^{2} \) |
| 79 | \( 1 + (0.730 + 1.15i)T + (-33.6 + 71.4i)T^{2} \) |
| 83 | \( 1 + (-3.20 + 1.26i)T + (60.5 - 56.8i)T^{2} \) |
| 89 | \( 1 + (5.81 - 2.30i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (10.3 + 12.4i)T + (-18.1 + 95.2i)T^{2} \) |
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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
−13.90569565505416982521077482009, −12.03692764387526908310495309822, −10.86277911479150877896875244513, −10.38366604954913003684180630208, −8.971664599361189162596287682363, −8.544334045033606522769175213504, −6.53573778240482124521981936887, −5.50454722755072139336728696966, −4.02785007633057687283214778558, −3.13927964180426165878500070345,
1.46602077778922149965251176102, 3.57405327020216918634164525651, 4.48835289746375710633172303799, 6.74710987082638202985737195154, 7.67385440490763658768584970055, 8.257108987900897423968587705989, 9.309883550825542700798391910272, 11.22744090053441419119366283879, 12.18035754115666662931279872824, 12.82778030569264943014006221030
