| L(s) = 1 | − 0.712i·3-s − 7.86·7-s + 8.49·9-s + (2.47 + 10.7i)11-s + 1.76·13-s − 15.1·17-s − 3.61i·19-s + 5.60i·21-s − 10.7i·23-s − 12.4i·27-s − 26.5i·29-s − 21.3·31-s + (7.64 − 1.76i)33-s + 1.48i·37-s − 1.25i·39-s + ⋯ |
| L(s) = 1 | − 0.237i·3-s − 1.12·7-s + 0.943·9-s + (0.225 + 0.974i)11-s + 0.135·13-s − 0.888·17-s − 0.190i·19-s + 0.267i·21-s − 0.465i·23-s − 0.461i·27-s − 0.916i·29-s − 0.690·31-s + (0.231 − 0.0535i)33-s + 0.0400i·37-s − 0.0322i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1227374011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1227374011\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-2.47 - 10.7i)T \) |
| good | 3 | \( 1 + 0.712iT - 9T^{2} \) |
| 7 | \( 1 + 7.86T + 49T^{2} \) |
| 13 | \( 1 - 1.76T + 169T^{2} \) |
| 17 | \( 1 + 15.1T + 289T^{2} \) |
| 19 | \( 1 + 3.61iT - 361T^{2} \) |
| 23 | \( 1 + 10.7iT - 529T^{2} \) |
| 29 | \( 1 + 26.5iT - 841T^{2} \) |
| 31 | \( 1 + 21.3T + 961T^{2} \) |
| 37 | \( 1 - 1.48iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 23.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 60.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 1.97iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 78.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 27.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 56.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 41.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 106.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 89.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 132.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 59.5T + 7.92e3T^{2} \) |
| 97 | \( 1 - 46.9iT - 9.40e3T^{2} \) |
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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
−9.879117908257898615363895358522, −9.438573625847960018651940298203, −8.431167791164064824079445113530, −7.30582415297790257497533859953, −6.79428485519166859324633705752, −6.04709984833699199082309588937, −4.69021691766850193848890012016, −3.98043141124842968024970048230, −2.73267695399468170023152774354, −1.57878531539570574614310023928,
0.03642218898494547316796299115, 1.59331659660442534811703676414, 3.13794414502331902529303886525, 3.79697940277321314560022370158, 4.90206203096446353240279223151, 6.01380910733432616823272652599, 6.70132262740849126491018089076, 7.50160361269594225663518816988, 8.703834404155646004479123669269, 9.259512990656722807047570256311
