L(s) = 1 | + 2·5-s + 4·7-s + 2·11-s − 2·13-s + 2·17-s + 2·19-s + 4·23-s − 25-s − 6·29-s + 8·35-s − 10·37-s + 6·41-s + 6·43-s − 8·47-s + 9·49-s − 6·53-s + 4·55-s − 14·59-s − 2·61-s − 4·65-s + 10·67-s + 12·71-s + 14·73-s + 8·77-s + 8·79-s + 6·83-s + 4·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.51·7-s + 0.603·11-s − 0.554·13-s + 0.485·17-s + 0.458·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 1.35·35-s − 1.64·37-s + 0.937·41-s + 0.914·43-s − 1.16·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s − 1.82·59-s − 0.256·61-s − 0.496·65-s + 1.22·67-s + 1.42·71-s + 1.63·73-s + 0.911·77-s + 0.900·79-s + 0.658·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.330452222\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.330452222\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.607232368240438516824809967267, −9.174652574334721806401375408825, −8.075661729852922503369502159976, −7.44440089656946414681700247943, −6.40345159211036072533921924868, −5.36530815656846370209456927790, −4.88235750431299832390164653972, −3.62014189552160099547672624510, −2.20034159198920039710092677140, −1.34910975935301153966920068234,
1.34910975935301153966920068234, 2.20034159198920039710092677140, 3.62014189552160099547672624510, 4.88235750431299832390164653972, 5.36530815656846370209456927790, 6.40345159211036072533921924868, 7.44440089656946414681700247943, 8.075661729852922503369502159976, 9.174652574334721806401375408825, 9.607232368240438516824809967267