| L(s) = 1 | − 3·5-s − 7-s + 6·11-s + 2·13-s + 6·17-s + 7·19-s − 3·23-s + 4·25-s + 6·29-s − 2·31-s + 3·35-s + 2·37-s − 2·43-s + 49-s + 6·53-s − 18·55-s + 5·61-s − 6·65-s − 8·67-s − 3·71-s + 2·73-s − 6·77-s − 5·79-s − 12·83-s − 18·85-s − 2·91-s − 21·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 1.80·11-s + 0.554·13-s + 1.45·17-s + 1.60·19-s − 0.625·23-s + 4/5·25-s + 1.11·29-s − 0.359·31-s + 0.507·35-s + 0.328·37-s − 0.304·43-s + 1/7·49-s + 0.824·53-s − 2.42·55-s + 0.640·61-s − 0.744·65-s − 0.977·67-s − 0.356·71-s + 0.234·73-s − 0.683·77-s − 0.562·79-s − 1.31·83-s − 1.95·85-s − 0.209·91-s − 2.15·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.981759946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.981759946\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.53392405688040166234159027223, −7.28398655102563440665641155725, −6.36936308079886949552594255205, −5.80022422799497912668542765275, −4.84530785618794272771352132490, −3.95032458644166792754612286528, −3.61554326376982968964547278915, −2.96538386588835707128705408626, −1.41909454250195247681572008309, −0.76917079518331910261210290751,
0.76917079518331910261210290751, 1.41909454250195247681572008309, 2.96538386588835707128705408626, 3.61554326376982968964547278915, 3.95032458644166792754612286528, 4.84530785618794272771352132490, 5.80022422799497912668542765275, 6.36936308079886949552594255205, 7.28398655102563440665641155725, 7.53392405688040166234159027223
