L(s) = 1 | + 6i·5-s − 5i·7-s − 6·11-s + 5.19i·13-s − 31.1·17-s + 25.9·19-s − 31.1i·23-s − 11·25-s − 36i·29-s + 26i·31-s + 30·35-s − 25.9i·37-s + 20.7·43-s − 31.1i·47-s + 24·49-s + ⋯ |
L(s) = 1 | + 1.20i·5-s − 0.714i·7-s − 0.545·11-s + 0.399i·13-s − 1.83·17-s + 1.36·19-s − 1.35i·23-s − 0.440·25-s − 1.24i·29-s + 0.838i·31-s + 0.857·35-s − 0.702i·37-s + 0.483·43-s − 0.663i·47-s + 0.489·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.434629535\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.434629535\) |
\(L(2)\) |
|
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 | \( 1 - 6iT - 25T^{2} \) |
| 7 | \( 1 + 5iT - 49T^{2} \) |
| 11 | \( 1 + 6T + 121T^{2} \) |
| 13 | \( 1 - 5.19iT - 169T^{2} \) |
| 17 | \( 1 + 31.1T + 289T^{2} \) |
| 19 | \( 1 - 25.9T + 361T^{2} \) |
| 23 | \( 1 + 31.1iT - 529T^{2} \) |
| 29 | \( 1 + 36iT - 841T^{2} \) |
| 31 | \( 1 - 26iT - 961T^{2} \) |
| 37 | \( 1 + 25.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 60iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 18T + 3.48e3T^{2} \) |
| 61 | \( 1 + 77.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 77.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 25T + 5.32e3T^{2} \) |
| 79 | \( 1 + 31iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 120T + 6.88e3T^{2} \) |
| 89 | \( 1 - 155.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 85T + 9.40e3T^{2} \) |
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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
−9.091066038491201475108411565217, −8.156908731168361382542524609026, −7.21883232271638412618397595704, −6.83141463896487556081011113440, −6.00053446389298124461394930974, −4.78972685707122645381240121856, −3.99925808550952381398336226064, −2.92957461216666183253174228215, −2.14702583419387544047571454795, −0.44802356264953959867370377858,
0.960563459503952240440666354666, 2.13488086233835796885912202220, 3.23455318406256749742760258440, 4.44488614821087089234793691291, 5.22623545635728580617380448273, 5.71446948665206574661276680631, 6.90607904298310569345630551630, 7.79151570429301368465591914501, 8.575926622570264907792617595693, 9.158558161106532582734850602424