L(s) = 1 | + 5-s + 7-s − 3·9-s − 2·13-s + 2·17-s + 8·19-s + 8·23-s + 25-s + 6·29-s + 35-s − 2·37-s − 6·41-s + 8·43-s − 3·45-s − 8·47-s + 49-s − 2·53-s + 8·59-s − 2·61-s − 3·63-s − 2·65-s + 8·67-s + 10·73-s − 16·79-s + 9·81-s + 16·83-s + 2·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s − 0.554·13-s + 0.485·17-s + 1.83·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.169·35-s − 0.328·37-s − 0.937·41-s + 1.21·43-s − 0.447·45-s − 1.16·47-s + 1/7·49-s − 0.274·53-s + 1.04·59-s − 0.256·61-s − 0.377·63-s − 0.248·65-s + 0.977·67-s + 1.17·73-s − 1.80·79-s + 81-s + 1.75·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.786140316\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.786140316\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.746787390867382157792095646105, −9.078413504844681466308296779634, −8.216853824182561035527399920283, −7.37688841827822057083742918824, −6.48150021918063889828568858074, −5.32351074773150468226638572671, −5.01822558158376692210748421008, −3.38685094418802578312920911584, −2.59698061649295808977252198284, −1.07477712752770183220289078477,
1.07477712752770183220289078477, 2.59698061649295808977252198284, 3.38685094418802578312920911584, 5.01822558158376692210748421008, 5.32351074773150468226638572671, 6.48150021918063889828568858074, 7.37688841827822057083742918824, 8.216853824182561035527399920283, 9.078413504844681466308296779634, 9.746787390867382157792095646105