| L(s) = 1 | − 3.31·2-s + (1.65 + 2.5i)3-s + 7·4-s + (−5.5 − 8.29i)6-s − 9.94·8-s + (−3.5 + 8.29i)9-s + 16.5i·11-s + (11.6 + 17.5i)12-s + 10i·13-s + 5.00·16-s − 3.31·17-s + (11.6 − 27.4i)18-s − 7·19-s − 55.0i·22-s + 19.8·23-s + (−16.5 − 24.8i)24-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + (0.552 + 0.833i)3-s + 1.75·4-s + (−0.916 − 1.38i)6-s − 1.24·8-s + (−0.388 + 0.921i)9-s + 1.50i·11-s + (0.967 + 1.45i)12-s + 0.769i·13-s + 0.312·16-s − 0.195·17-s + (0.644 − 1.52i)18-s − 0.368·19-s − 2.50i·22-s + 0.865·23-s + (−0.687 − 1.03i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.423227 + 0.478306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.423227 + 0.478306i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.65 - 2.5i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.31T + 4T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 16.5iT - 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 + 3.31T + 289T^{2} \) |
| 19 | \( 1 + 7T + 361T^{2} \) |
| 23 | \( 1 - 19.8T + 529T^{2} \) |
| 29 | \( 1 + 33.1iT - 841T^{2} \) |
| 31 | \( 1 - 42T + 961T^{2} \) |
| 37 | \( 1 - 40iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 46.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 66.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 45iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 33.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 35iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 12T + 6.24e3T^{2} \) |
| 83 | \( 1 - 69.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 149. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 70iT - 9.40e3T^{2} \) |
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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
−15.13288639524818421349808910885, −13.69968483671011841339864106806, −11.95389046428366435455292877285, −10.72465697462911655549271894807, −9.840655984491762970981446057829, −9.117818490591680144961085547601, −8.028262890006306496922678387450, −6.85375806299344915528016160779, −4.51991831460946341657674476768, −2.22481081588170554204089271884,
0.908229494399085101493476265249, 2.88055083721362980639495000710, 6.14761488436847336722925105830, 7.39831720023179465024042714305, 8.414257734253821439877396532851, 9.050575231139934831003180986404, 10.50981825181986785872294517575, 11.46011071517585769267112723919, 12.85905828912534486385178688375, 13.98736952966368462425964918412
