| L(s) = 1 | − 5-s − 4·7-s − 3·9-s − 4·13-s + 4·17-s − 8·19-s + 4·23-s + 25-s + 8·29-s − 4·31-s + 4·35-s + 6·37-s + 8·41-s − 4·43-s + 3·45-s + 12·47-s + 9·49-s − 10·53-s − 8·61-s + 12·63-s + 4·65-s + 8·67-s + 12·71-s − 12·73-s + 8·79-s + 9·81-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 9-s − 1.10·13-s + 0.970·17-s − 1.83·19-s + 0.834·23-s + 1/5·25-s + 1.48·29-s − 0.718·31-s + 0.676·35-s + 0.986·37-s + 1.24·41-s − 0.609·43-s + 0.447·45-s + 1.75·47-s + 9/7·49-s − 1.37·53-s − 1.02·61-s + 1.51·63-s + 0.496·65-s + 0.977·67-s + 1.42·71-s − 1.40·73-s + 0.900·79-s + 81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7719498979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7719498979\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 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
−9.034834451058096715126853018952, −8.201585180178097915446335336855, −7.41467720526900238136829562725, −6.54090656858216715668913346690, −6.01499620829364709574074390058, −5.00415614765738623982384006360, −4.03973384262460596898424780787, −3.05788151016232052911799299337, −2.50546757714043366766933909453, −0.52619618336172832018243132876,
0.52619618336172832018243132876, 2.50546757714043366766933909453, 3.05788151016232052911799299337, 4.03973384262460596898424780787, 5.00415614765738623982384006360, 6.01499620829364709574074390058, 6.54090656858216715668913346690, 7.41467720526900238136829562725, 8.201585180178097915446335336855, 9.034834451058096715126853018952
