L(s) = 1 | − 2·5-s − 4·11-s + 2·13-s + 6·17-s + 19-s − 4·23-s − 25-s + 2·29-s − 4·31-s + 10·37-s − 10·41-s − 4·43-s − 4·47-s − 7·49-s + 10·53-s + 8·55-s + 12·59-s + 14·61-s − 4·65-s + 12·67-s + 8·71-s − 6·73-s + 4·79-s + 12·83-s − 12·85-s + 6·89-s − 2·95-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.229·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s + 1.64·37-s − 1.56·41-s − 0.609·43-s − 0.583·47-s − 49-s + 1.37·53-s + 1.07·55-s + 1.56·59-s + 1.79·61-s − 0.496·65-s + 1.46·67-s + 0.949·71-s − 0.702·73-s + 0.450·79-s + 1.31·83-s − 1.30·85-s + 0.635·89-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.281924272\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.281924272\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.501995439933935386642992901059, −8.057224083383321383561004935430, −7.54188937600423423101808663065, −6.59739773494044819386161896201, −5.60441811878021212455097225817, −5.02382748416547556621669886282, −3.86301997176052042051145556459, −3.33052988139656335920774502455, −2.15606001082909168122252072981, −0.69408070740984200675842868720,
0.69408070740984200675842868720, 2.15606001082909168122252072981, 3.33052988139656335920774502455, 3.86301997176052042051145556459, 5.02382748416547556621669886282, 5.60441811878021212455097225817, 6.59739773494044819386161896201, 7.54188937600423423101808663065, 8.057224083383321383561004935430, 8.501995439933935386642992901059