| L(s) = 1 | + 2·5-s − 7-s − 3·9-s − 4·11-s + 2·13-s − 6·17-s + 8·19-s − 25-s + 6·29-s + 8·31-s − 2·35-s − 2·37-s + 2·41-s − 4·43-s − 6·45-s − 8·47-s + 49-s + 6·53-s − 8·55-s − 6·61-s + 3·63-s + 4·65-s − 4·67-s − 8·71-s + 10·73-s + 4·77-s + 16·79-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 0.768·61-s + 0.377·63-s + 0.496·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8745483141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8745483141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.41323382474483111142553586928, −13.77746713200327219211224912285, −13.45094853660694360023256897823, −11.84146297682028690633889717654, −10.60224244605800499935670328540, −9.457349313423188294666539437989, −8.188420417684287628984093773695, −6.42262084813563967264199691637, −5.22316409435338364257345412913, −2.79183800612725741835263061431,
2.79183800612725741835263061431, 5.22316409435338364257345412913, 6.42262084813563967264199691637, 8.188420417684287628984093773695, 9.457349313423188294666539437989, 10.60224244605800499935670328540, 11.84146297682028690633889717654, 13.45094853660694360023256897823, 13.77746713200327219211224912285, 15.41323382474483111142553586928
