| L(s) = 1 | + (−0.151 − 2.01i)2-s + (1.33 + 0.412i)3-s + (−2.06 + 0.310i)4-s + (−1.66 + 1.54i)5-s + (0.629 − 2.76i)6-s + (−1.77 + 1.95i)7-s + (0.0388 + 0.170i)8-s + (−0.858 − 0.584i)9-s + (3.37 + 3.12i)10-s + (1.57 − 1.07i)11-s + (−2.88 − 0.435i)12-s + (4.29 + 2.06i)13-s + (4.21 + 3.28i)14-s + (−2.87 + 1.38i)15-s + (−3.65 + 1.12i)16-s + (−1.47 − 3.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.106 − 1.42i)2-s + (0.772 + 0.238i)3-s + (−1.03 + 0.155i)4-s + (−0.745 + 0.692i)5-s + (0.257 − 1.12i)6-s + (−0.671 + 0.740i)7-s + (0.0137 + 0.0601i)8-s + (−0.286 − 0.194i)9-s + (1.06 + 0.989i)10-s + (0.475 − 0.324i)11-s + (−0.834 − 0.125i)12-s + (1.19 + 0.573i)13-s + (1.12 + 0.878i)14-s + (−0.741 + 0.356i)15-s + (−0.912 + 0.281i)16-s + (−0.358 − 0.914i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.678476 - 0.494628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678476 - 0.494628i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.77 - 1.95i)T \) |
| good | 2 | \( 1 + (0.151 + 2.01i)T + (-1.97 + 0.298i)T^{2} \) |
| 3 | \( 1 + (-1.33 - 0.412i)T + (2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (1.66 - 1.54i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (-1.57 + 1.07i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-4.29 - 2.06i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (1.47 + 3.76i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-0.218 - 0.379i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.49 + 6.35i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (5.30 - 6.64i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (0.409 - 0.709i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.68 - 0.555i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-2.40 - 10.5i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-1.79 + 7.87i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.114 - 1.53i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (-0.818 + 0.123i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (2.23 + 2.07i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.0576 - 0.00869i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-6.06 + 10.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.05 - 3.83i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.809 - 10.8i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (1.22 + 2.13i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.16 - 2.00i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (3.35 + 2.29i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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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
−15.27678747534733236685282038185, −14.16515065600748207108430875633, −12.89019202858721180510896992985, −11.64622543064035633172985139689, −10.99492566767919878671855605569, −9.412697870493973541493776701858, −8.708948973887031283098563696316, −6.60715651189536880904590832936, −3.75844927420940842911488618835, −2.83692731735832490189411058095,
3.93237061296174293806989214568, 5.91494540484763681950985226682, 7.41274009356171215653093201065, 8.230176685988074165768637026139, 9.226964059605660142590728131912, 11.21732687306566565305305467948, 13.00829519132183444950190829452, 13.76713617058722842780206151260, 15.03629688033677944972190542769, 15.79744875144760282794257693394
