| L(s) = 1 | + 5-s + 3·7-s + 6·11-s − 2·13-s − 3·17-s − 6·19-s + 23-s + 25-s + 9·29-s − 3·31-s + 3·35-s + 3·37-s + 3·41-s − 4·47-s + 2·49-s + 9·53-s + 6·55-s + 3·59-s − 8·61-s − 2·65-s + 3·67-s + 9·71-s + 6·73-s + 18·77-s − 4·79-s − 3·83-s − 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 1.80·11-s − 0.554·13-s − 0.727·17-s − 1.37·19-s + 0.208·23-s + 1/5·25-s + 1.67·29-s − 0.538·31-s + 0.507·35-s + 0.493·37-s + 0.468·41-s − 0.583·47-s + 2/7·49-s + 1.23·53-s + 0.809·55-s + 0.390·59-s − 1.02·61-s − 0.248·65-s + 0.366·67-s + 1.06·71-s + 0.702·73-s + 2.05·77-s − 0.450·79-s − 0.329·83-s − 0.325·85-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
| \(L(1)\) |
\(\approx\) |
\(2.926518441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.926518441\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.87880322153703235580423716239, −6.93593278211091926304418793179, −6.51913664445892458637730073172, −5.83616943972171789143694628040, −4.74213422371018555798400799261, −4.49862202267222825947986074707, −3.62403715120826288083600251024, −2.41137123841173901822357596523, −1.80866430095552834503618232943, −0.882408719248005362401823484559,
0.882408719248005362401823484559, 1.80866430095552834503618232943, 2.41137123841173901822357596523, 3.62403715120826288083600251024, 4.49862202267222825947986074707, 4.74213422371018555798400799261, 5.83616943972171789143694628040, 6.51913664445892458637730073172, 6.93593278211091926304418793179, 7.87880322153703235580423716239
