L(s) = 1 | + 5-s + 3·7-s + 2·11-s + 2·13-s + 17-s + 6·19-s + 23-s + 25-s − 3·29-s − 3·31-s + 3·35-s + 3·37-s + 7·41-s + 2·49-s + 5·53-s + 2·55-s − 9·59-s + 12·61-s + 2·65-s + 3·67-s − 3·71-s + 2·73-s + 6·77-s + 4·79-s − 7·83-s + 85-s + 16·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.13·7-s + 0.603·11-s + 0.554·13-s + 0.242·17-s + 1.37·19-s + 0.208·23-s + 1/5·25-s − 0.557·29-s − 0.538·31-s + 0.507·35-s + 0.493·37-s + 1.09·41-s + 2/7·49-s + 0.686·53-s + 0.269·55-s − 1.17·59-s + 1.53·61-s + 0.248·65-s + 0.366·67-s − 0.356·71-s + 0.234·73-s + 0.683·77-s + 0.450·79-s − 0.768·83-s + 0.108·85-s + 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.222263603\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.222263603\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.80199992552741882263747836141, −7.21009139979540481553797900374, −6.39887133746837743409179918956, −5.59114036299381395365920888935, −5.17495333875979440729876994762, −4.26485095427431206296578491959, −3.56683091876985132372764754871, −2.58682631519063835259978211132, −1.61642574199861404169485097873, −0.984899066578597768736073795862,
0.984899066578597768736073795862, 1.61642574199861404169485097873, 2.58682631519063835259978211132, 3.56683091876985132372764754871, 4.26485095427431206296578491959, 5.17495333875979440729876994762, 5.59114036299381395365920888935, 6.39887133746837743409179918956, 7.21009139979540481553797900374, 7.80199992552741882263747836141