L(s) = 1 | + 1.13i·2-s + i·3-s + 0.714·4-s − 2.77i·5-s − 1.13·6-s + (2.60 + 0.474i)7-s + 3.07i·8-s − 9-s + 3.14·10-s + (3.23 + 0.726i)11-s + 0.714i·12-s − 1.20·13-s + (−0.538 + 2.95i)14-s + 2.77·15-s − 2.05·16-s − 6.01·17-s + ⋯ |
L(s) = 1 | + 0.801i·2-s + 0.577i·3-s + 0.357·4-s − 1.24i·5-s − 0.462·6-s + (0.983 + 0.179i)7-s + 1.08i·8-s − 0.333·9-s + 0.994·10-s + (0.975 + 0.219i)11-s + 0.206i·12-s − 0.333·13-s + (−0.143 + 0.788i)14-s + 0.716·15-s − 0.514·16-s − 1.46·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25105 + 0.828143i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25105 + 0.828143i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (-2.60 - 0.474i)T \) |
| 11 | \( 1 + (-3.23 - 0.726i)T \) |
good | 2 | \( 1 - 1.13iT - 2T^{2} \) |
| 5 | \( 1 + 2.77iT - 5T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 17 | \( 1 + 6.01T + 17T^{2} \) |
| 19 | \( 1 - 2.01T + 19T^{2} \) |
| 23 | \( 1 + 5.90T + 23T^{2} \) |
| 29 | \( 1 + 6.40iT - 29T^{2} \) |
| 31 | \( 1 - 6.11iT - 31T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 + 0.503T + 41T^{2} \) |
| 43 | \( 1 + 9.37iT - 43T^{2} \) |
| 47 | \( 1 + 7.12iT - 47T^{2} \) |
| 53 | \( 1 + 1.42T + 53T^{2} \) |
| 59 | \( 1 - 5.81iT - 59T^{2} \) |
| 61 | \( 1 + 4.53T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 16.3iT - 79T^{2} \) |
| 83 | \( 1 - 0.143T + 83T^{2} \) |
| 89 | \( 1 - 3.07iT - 89T^{2} \) |
| 97 | \( 1 + 3.26iT - 97T^{2} \) |
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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
−12.00828746168160763980917677305, −11.69409618407556311330326704750, −10.44351608837641188526671093955, −9.043817957172393858999028754231, −8.546434828944333073925903668943, −7.45846375264937123795655523474, −6.15868321609995678660980151450, −5.07521011065438507376142084777, −4.29359518060411707771612763796, −1.95754150913041106278678099299,
1.69582882614104178319353524381, 2.83851538455157053922643525821, 4.21119746706279844174673139845, 6.19307999496505580221740165114, 6.94168101240381606441479888816, 7.85275074547726898052812358626, 9.302772965616833383555688876758, 10.48903786632577293826534266509, 11.30287103274358023357527037172, 11.60864195567320570279039000195