| L(s) = 1 | + 0.306·3-s + 2.60·7-s − 2.90·9-s − 2.71·11-s + 0.503·13-s + 5.94·17-s + 5.72·19-s + 0.799·21-s − 1.28·23-s − 1.81·27-s + 2.11·29-s − 3.95·31-s − 0.831·33-s + 0.825·37-s + 0.154·39-s − 4.53·41-s + 5.38·43-s − 5.62·47-s − 0.205·49-s + 1.82·51-s + 10.9·53-s + 1.75·57-s + 13.7·59-s − 7.00·61-s − 7.57·63-s − 2.85·67-s − 0.394·69-s + ⋯ |
| L(s) = 1 | + 0.176·3-s + 0.985·7-s − 0.968·9-s − 0.818·11-s + 0.139·13-s + 1.44·17-s + 1.31·19-s + 0.174·21-s − 0.268·23-s − 0.348·27-s + 0.392·29-s − 0.710·31-s − 0.144·33-s + 0.135·37-s + 0.0246·39-s − 0.707·41-s + 0.821·43-s − 0.819·47-s − 0.0294·49-s + 0.255·51-s + 1.50·53-s + 0.232·57-s + 1.79·59-s − 0.896·61-s − 0.954·63-s − 0.348·67-s − 0.0475·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131582554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131582554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.306T + 3T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 - 0.503T + 13T^{2} \) |
| 17 | \( 1 - 5.94T + 17T^{2} \) |
| 19 | \( 1 - 5.72T + 19T^{2} \) |
| 23 | \( 1 + 1.28T + 23T^{2} \) |
| 29 | \( 1 - 2.11T + 29T^{2} \) |
| 31 | \( 1 + 3.95T + 31T^{2} \) |
| 37 | \( 1 - 0.825T + 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 + 5.62T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 7.00T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 7.02T + 97T^{2} \) |
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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
−8.315188795635865697496631188165, −7.80428423455454623657033324018, −7.26532758660719514118034074379, −6.02453108061362074549569151098, −5.37887264519669949928535237885, −4.94464768356629311375536181624, −3.67835338587488315604894675673, −2.99364318037871189991859861643, −2.01528364310554876741517759295, −0.843926610585343151757524194620,
0.843926610585343151757524194620, 2.01528364310554876741517759295, 2.99364318037871189991859861643, 3.67835338587488315604894675673, 4.94464768356629311375536181624, 5.37887264519669949928535237885, 6.02453108061362074549569151098, 7.26532758660719514118034074379, 7.80428423455454623657033324018, 8.315188795635865697496631188165
