| L(s) = 1 | + 2.82·5-s + 4·11-s + 2.82·13-s − 5.65·17-s + 2.82·19-s + 3.00·25-s − 2·29-s + 5.65·31-s + 10·37-s + 5.65·41-s − 4·43-s + 5.65·47-s − 6·53-s + 11.3·55-s − 2.82·59-s − 14.1·61-s + 8.00·65-s + 12·67-s + 8·79-s − 14.1·83-s − 16.0·85-s + 8.00·95-s − 5.65·97-s − 8.48·101-s + 16.9·103-s + 12·107-s + 2·109-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + 1.20·11-s + 0.784·13-s − 1.37·17-s + 0.648·19-s + 0.600·25-s − 0.371·29-s + 1.01·31-s + 1.64·37-s + 0.883·41-s − 0.609·43-s + 0.825·47-s − 0.824·53-s + 1.52·55-s − 0.368·59-s − 1.81·61-s + 0.992·65-s + 1.46·67-s + 0.900·79-s − 1.55·83-s − 1.73·85-s + 0.820·95-s − 0.574·97-s − 0.844·101-s + 1.67·103-s + 1.16·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.771219980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.771219980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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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
−8.779091007299087085196726220357, −7.88493473485601880256223068312, −6.85057443497995944470937656473, −6.24954272416215348011693505943, −5.82589950964702926081659478904, −4.71561811968611740316850013120, −4.00417021186945233057196570413, −2.88551589580167821469342231874, −1.94223960823426379351982495343, −1.06134693259088244155334188862,
1.06134693259088244155334188862, 1.94223960823426379351982495343, 2.88551589580167821469342231874, 4.00417021186945233057196570413, 4.71561811968611740316850013120, 5.82589950964702926081659478904, 6.24954272416215348011693505943, 6.85057443497995944470937656473, 7.88493473485601880256223068312, 8.779091007299087085196726220357
