L(s) = 1 | − 2·5-s + 4·7-s − 3·9-s + 11-s + 2·13-s + 2·17-s − 19-s − 25-s + 2·29-s − 8·35-s + 10·37-s − 2·41-s + 4·43-s + 6·45-s − 8·47-s + 9·49-s + 2·53-s − 2·55-s − 8·59-s + 10·61-s − 12·63-s − 4·65-s + 8·67-s + 2·73-s + 4·77-s − 8·79-s + 9·81-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.51·7-s − 9-s + 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.229·19-s − 1/5·25-s + 0.371·29-s − 1.35·35-s + 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.269·55-s − 1.04·59-s + 1.28·61-s − 1.51·63-s − 0.496·65-s + 0.977·67-s + 0.234·73-s + 0.455·77-s − 0.900·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.768748692\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.768748692\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.308479977864138050186821633083, −8.110615557594120688597690938236, −7.37173009338229018797977934191, −6.30617915603259027650428119557, −5.55252556811006636089894948473, −4.71605349876078591156289230358, −4.02416329474396950777130627788, −3.11572598713517635695637220431, −1.98469055489481903292064347062, −0.811372830191678156603204242051,
0.811372830191678156603204242051, 1.98469055489481903292064347062, 3.11572598713517635695637220431, 4.02416329474396950777130627788, 4.71605349876078591156289230358, 5.55252556811006636089894948473, 6.30617915603259027650428119557, 7.37173009338229018797977934191, 8.110615557594120688597690938236, 8.308479977864138050186821633083