L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 6·11-s + 4·13-s − 2·14-s + 16-s + 6·17-s − 4·19-s + 6·22-s + 4·26-s − 2·28-s − 6·29-s − 4·31-s + 32-s + 6·34-s − 8·37-s − 4·38-s − 8·43-s + 6·44-s − 3·49-s + 4·52-s + 6·53-s − 2·56-s − 6·58-s + 6·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 1.80·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 1.27·22-s + 0.784·26-s − 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 1.31·37-s − 0.648·38-s − 1.21·43-s + 0.904·44-s − 3/7·49-s + 0.554·52-s + 0.824·53-s − 0.267·56-s − 0.787·58-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.200233084\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.200233084\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−11.30176306918800542324895517156, −10.26974685380686003817751853982, −9.322316995793925649964979530065, −8.428134066544604724277886211493, −7.05855949257969394331275338730, −6.35337453909666730560193822981, −5.50042078426760473642213736203, −3.93114117360591390680316014330, −3.44283204101776149845879223587, −1.55715684978414451776466943001,
1.55715684978414451776466943001, 3.44283204101776149845879223587, 3.93114117360591390680316014330, 5.50042078426760473642213736203, 6.35337453909666730560193822981, 7.05855949257969394331275338730, 8.428134066544604724277886211493, 9.322316995793925649964979530065, 10.26974685380686003817751853982, 11.30176306918800542324895517156