| L(s) = 1 | − 3-s − 4·7-s + 9-s + 6·13-s + 2·17-s + 4·19-s + 4·21-s + 8·23-s − 27-s − 6·29-s + 6·37-s − 6·39-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s − 2·51-s − 10·53-s − 4·57-s + 6·61-s − 4·63-s + 4·67-s − 8·69-s + 14·73-s + 16·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.66·13-s + 0.485·17-s + 0.917·19-s + 0.872·21-s + 1.66·23-s − 0.192·27-s − 1.11·29-s + 0.986·37-s − 0.960·39-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s − 0.529·57-s + 0.768·61-s − 0.503·63-s + 0.488·67-s − 0.963·69-s + 1.63·73-s + 1.80·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.113481484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113481484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.88178013654136620600664945750, −9.615274856932354829752196884026, −9.267882456759456014915649652272, −7.941429860491494173937375470302, −6.86451500512904244632587384877, −6.16158934355050204830273528878, −5.35537149321682671303684629035, −3.87279651685007961866456189265, −3.04390454355544402274508317221, −0.989067407202575093989545666411,
0.989067407202575093989545666411, 3.04390454355544402274508317221, 3.87279651685007961866456189265, 5.35537149321682671303684629035, 6.16158934355050204830273528878, 6.86451500512904244632587384877, 7.941429860491494173937375470302, 9.267882456759456014915649652272, 9.615274856932354829752196884026, 10.88178013654136620600664945750
