| L(s) = 1 | + (−1.29 + 0.578i)2-s + (−1.32 − 0.887i)3-s + (1.33 − 1.49i)4-s + (−2.01 + 0.971i)5-s + (2.22 + 0.377i)6-s + (2.49 + 1.03i)7-s + (−0.856 + 2.69i)8-s + (−0.172 − 0.415i)9-s + (2.03 − 2.41i)10-s + (0.425 − 0.284i)11-s + (−3.09 + 0.799i)12-s + (2.48 − 0.494i)13-s + (−3.82 + 0.108i)14-s + (3.53 + 0.496i)15-s + (−0.452 − 3.97i)16-s + (−4.97 − 4.97i)17-s + ⋯ |
| L(s) = 1 | + (−0.912 + 0.408i)2-s + (−0.766 − 0.512i)3-s + (0.665 − 0.746i)4-s + (−0.900 + 0.434i)5-s + (0.908 + 0.154i)6-s + (0.944 + 0.391i)7-s + (−0.302 + 0.953i)8-s + (−0.0573 − 0.138i)9-s + (0.644 − 0.764i)10-s + (0.128 − 0.0856i)11-s + (−0.892 + 0.230i)12-s + (0.688 − 0.137i)13-s + (−1.02 + 0.0289i)14-s + (0.913 + 0.128i)15-s + (−0.113 − 0.993i)16-s + (−1.20 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.163112 - 0.262949i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.163112 - 0.262949i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.29 - 0.578i)T \) |
| 5 | \( 1 + (2.01 - 0.971i)T \) |
| good | 3 | \( 1 + (1.32 + 0.887i)T + (1.14 + 2.77i)T^{2} \) |
| 7 | \( 1 + (-2.49 - 1.03i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.425 + 0.284i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-2.48 + 0.494i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (4.97 + 4.97i)T + 17iT^{2} \) |
| 19 | \( 1 + (6.54 - 1.30i)T + (17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (1.66 + 4.02i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (7.69 + 5.14i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + 3.69iT - 31T^{2} \) |
| 37 | \( 1 + (-1.07 + 5.42i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-7.15 + 2.96i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (2.99 - 1.99i)T + (16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (-6.48 - 6.48i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.01 + 7.49i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-0.192 + 0.967i)T + (-54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (0.973 - 1.45i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (0.735 + 0.491i)T + (25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (2.68 - 6.48i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (4.55 - 1.88i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (4.36 + 4.36i)T + 79iT^{2} \) |
| 83 | \( 1 + (-0.448 - 2.25i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (-12.3 - 5.11i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−11.23467519484144649224382108343, −10.79327776693350726755323422526, −9.208991164481432673856804866214, −8.386897611174768242381649612076, −7.51936097056094503118412481126, −6.57290621515754181941157422201, −5.80003329385247943473337539379, −4.33487159269910794757526044661, −2.23483404678606474620822768879, −0.32317223265952958196746858908,
1.69656407378154576071179480321, 3.86936517992214160470875135781, 4.62550372078086075422037187068, 6.17065742884744499217855658577, 7.43289182414843481935106209861, 8.366017391964060038827251321806, 8.956025621871484300819704363156, 10.48585125817858105041300104764, 11.02692026010645265419128897337, 11.40776146295088572692051083021
