| L(s) = 1 | + 5-s + 3·7-s + 2·11-s − 13-s + 3·17-s − 2·19-s − 4·23-s − 4·25-s + 2·29-s + 4·31-s + 3·35-s − 5·37-s + 12·41-s − 7·43-s + 9·47-s + 2·49-s + 4·53-s + 2·55-s + 6·59-s + 4·61-s − 65-s + 10·67-s + 15·71-s − 2·73-s + 6·77-s − 8·79-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 0.603·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s − 0.834·23-s − 4/5·25-s + 0.371·29-s + 0.718·31-s + 0.507·35-s − 0.821·37-s + 1.87·41-s − 1.06·43-s + 1.31·47-s + 2/7·49-s + 0.549·53-s + 0.269·55-s + 0.781·59-s + 0.512·61-s − 0.124·65-s + 1.22·67-s + 1.78·71-s − 0.234·73-s + 0.683·77-s − 0.900·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.772193302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.772193302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.964703245594772929984299285530, −7.26452379598760388338469331118, −6.44784880927663012596041055664, −5.75101064463899872805706654704, −5.13453009003348958858406129103, −4.31169037721870521245593585722, −3.69882720223680610508104183144, −2.47414386983859427575410557935, −1.83509402100962992787287050734, −0.870411349685264859192266368730,
0.870411349685264859192266368730, 1.83509402100962992787287050734, 2.47414386983859427575410557935, 3.69882720223680610508104183144, 4.31169037721870521245593585722, 5.13453009003348958858406129103, 5.75101064463899872805706654704, 6.44784880927663012596041055664, 7.26452379598760388338469331118, 7.964703245594772929984299285530
