| L(s) = 1 | + (−0.615 + 1.27i)2-s + (−0.519 − 0.176i)3-s + (−1.24 − 1.56i)4-s + (0.543 − 1.10i)5-s + (0.544 − 0.552i)6-s + (−2.35 + 1.20i)7-s + (2.76 − 0.614i)8-s + (−2.14 − 1.64i)9-s + (1.06 + 1.37i)10-s + (3.68 + 3.23i)11-s + (0.368 + 1.03i)12-s + (−1.84 + 2.76i)13-s + (−0.0838 − 3.74i)14-s + (−0.476 + 0.476i)15-s + (−0.917 + 3.89i)16-s + (5.32 − 1.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.435 + 0.900i)2-s + (−0.299 − 0.101i)3-s + (−0.620 − 0.784i)4-s + (0.243 − 0.493i)5-s + (0.222 − 0.225i)6-s + (−0.890 + 0.455i)7-s + (0.976 − 0.217i)8-s + (−0.713 − 0.547i)9-s + (0.338 + 0.433i)10-s + (1.11 + 0.974i)11-s + (0.106 + 0.298i)12-s + (−0.513 + 0.767i)13-s + (−0.0224 − 0.999i)14-s + (−0.123 + 0.123i)15-s + (−0.229 + 0.973i)16-s + (1.29 − 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.679962 + 0.571431i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679962 + 0.571431i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.615 - 1.27i)T \) |
| 7 | \( 1 + (2.35 - 1.20i)T \) |
| good | 3 | \( 1 + (0.519 + 0.176i)T + (2.38 + 1.82i)T^{2} \) |
| 5 | \( 1 + (-0.543 + 1.10i)T + (-3.04 - 3.96i)T^{2} \) |
| 11 | \( 1 + (-3.68 - 3.23i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (1.84 - 2.76i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-5.32 + 1.42i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.39 - 0.353i)T + (18.8 + 2.47i)T^{2} \) |
| 23 | \( 1 + (-5.75 - 4.41i)T + (5.95 + 22.2i)T^{2} \) |
| 29 | \( 1 + (2.25 - 0.448i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-1.12 - 0.646i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.36 + 4.80i)T + (-22.5 - 29.3i)T^{2} \) |
| 41 | \( 1 + (10.7 - 4.46i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (1.05 + 0.208i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-5.07 - 1.35i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.35 - 3.82i)T + (-6.91 - 52.5i)T^{2} \) |
| 59 | \( 1 + (0.898 + 13.7i)T + (-58.4 + 7.70i)T^{2} \) |
| 61 | \( 1 + (0.106 + 0.121i)T + (-7.96 + 60.4i)T^{2} \) |
| 67 | \( 1 + (2.31 - 6.82i)T + (-53.1 - 40.7i)T^{2} \) |
| 71 | \( 1 + (5.34 - 12.9i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-10.6 - 1.40i)T + (70.5 + 18.8i)T^{2} \) |
| 79 | \( 1 + (2.70 + 0.724i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-1.14 + 1.71i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (0.0938 + 0.712i)T + (-85.9 + 23.0i)T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−11.44131519328117352081933014384, −9.737392787840531327858116192722, −9.505575868028098209244552193065, −8.826541073340273145350145902611, −7.35307625315570839369320602307, −6.76459892610168530244648189938, −5.70124090437350368219676120953, −5.00213050116409963356662186901, −3.42318064570415760504692879744, −1.26472368303630195600223866937,
0.816208967149812295638730068919, 2.89601468244070386895738653016, 3.45610940167272554638990222883, 5.05585975651502066650767889375, 6.19173961751314530748345457623, 7.32983128187679733233968732734, 8.377161372363088086308590016605, 9.319398324351472431914484657890, 10.25490113103931768235402384334, 10.69326321601279089288383468956
