L(s) = 1 | − 5.19i·5-s − 6.34·7-s + 9.43i·11-s + 19.6·13-s − 1.90i·17-s − 4.69·19-s − 9.43i·23-s − 2·25-s + 3.28i·29-s + 41.0·31-s + 32.9i·35-s + 17.3·37-s − 61.8i·41-s − 0.954·43-s − 14.1i·47-s + ⋯ |
L(s) = 1 | − 1.03i·5-s − 0.906·7-s + 0.858i·11-s + 1.51·13-s − 0.112i·17-s − 0.247·19-s − 0.410i·23-s − 0.0800·25-s + 0.113i·29-s + 1.32·31-s + 0.942i·35-s + 0.467·37-s − 1.50i·41-s − 0.0221·43-s − 0.300i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.549007605\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.549007605\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 5.19iT - 25T^{2} \) |
| 7 | \( 1 + 6.34T + 49T^{2} \) |
| 11 | \( 1 - 9.43iT - 121T^{2} \) |
| 13 | \( 1 - 19.6T + 169T^{2} \) |
| 17 | \( 1 + 1.90iT - 289T^{2} \) |
| 19 | \( 1 + 4.69T + 361T^{2} \) |
| 23 | \( 1 + 9.43iT - 529T^{2} \) |
| 29 | \( 1 - 3.28iT - 841T^{2} \) |
| 31 | \( 1 - 41.0T + 961T^{2} \) |
| 37 | \( 1 - 17.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 61.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 0.954T + 1.84e3T^{2} \) |
| 47 | \( 1 + 14.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 91.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 75.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 30.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 29.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 87.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 95.8T + 9.40e3T^{2} \) |
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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
−9.111891606606654420001004133416, −8.660748267562418404880884295884, −7.72920429256583852277171081834, −6.63674795616273895876505659725, −6.02930158113953161419912389683, −4.92721674740632177856721791498, −4.15019956964612132070685409468, −3.12099422938010203503235940134, −1.71761702231169970825154405321, −0.51741568217380455299230901792,
1.12732748443671279187222178068, 2.82858162081145648568382306321, 3.32062594632506754131328687417, 4.35137207489972473608390977251, 5.96940706798490210695117946167, 6.19606488363671555771599339384, 7.05932283752944679769510658261, 8.112350102504115131242659510604, 8.818909936611294275325454614241, 9.772353217227276549279935759486