| L(s) = 1 | − 5-s − 3·7-s − 3·11-s − 6·13-s − 3·17-s + 19-s + 4·23-s − 4·25-s + 10·29-s − 2·31-s + 3·35-s + 8·37-s + 8·41-s + 43-s + 3·47-s + 2·49-s + 6·53-s + 3·55-s + 7·61-s + 6·65-s − 8·67-s + 12·71-s − 11·73-s + 9·77-s + 4·83-s + 3·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 0.904·11-s − 1.66·13-s − 0.727·17-s + 0.229·19-s + 0.834·23-s − 4/5·25-s + 1.85·29-s − 0.359·31-s + 0.507·35-s + 1.31·37-s + 1.24·41-s + 0.152·43-s + 0.437·47-s + 2/7·49-s + 0.824·53-s + 0.404·55-s + 0.896·61-s + 0.744·65-s − 0.977·67-s + 1.42·71-s − 1.28·73-s + 1.02·77-s + 0.439·83-s + 0.325·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8465536765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8465536765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.892718237653967146997373477022, −7.935555959343104225865564799258, −7.32016227740693371415400237232, −6.66260307528373962910372616260, −5.74803450606645387288578022188, −4.85468716698404797055779902636, −4.12511627512637579479297949689, −2.90182171573806321583393234308, −2.47342224903748840497823197638, −0.53708427580882030700657091576,
0.53708427580882030700657091576, 2.47342224903748840497823197638, 2.90182171573806321583393234308, 4.12511627512637579479297949689, 4.85468716698404797055779902636, 5.74803450606645387288578022188, 6.66260307528373962910372616260, 7.32016227740693371415400237232, 7.935555959343104225865564799258, 8.892718237653967146997373477022
