| L(s) = 1 | − 5·5-s + 36·7-s + 64·11-s − 65·13-s + 59·17-s − 28·19-s + 160·23-s − 100·25-s − 57·29-s + 164·31-s − 180·35-s − 321·37-s − 246·41-s − 8·43-s + 84·47-s + 953·49-s + 478·53-s − 320·55-s − 32·59-s + 415·61-s + 325·65-s − 220·67-s + 884·71-s − 77·73-s + 2.30e3·77-s − 80·79-s + 1.26e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.94·7-s + 1.75·11-s − 1.38·13-s + 0.841·17-s − 0.338·19-s + 1.45·23-s − 4/5·25-s − 0.364·29-s + 0.950·31-s − 0.869·35-s − 1.42·37-s − 0.937·41-s − 0.0283·43-s + 0.260·47-s + 2.77·49-s + 1.23·53-s − 0.784·55-s − 0.0706·59-s + 0.871·61-s + 0.620·65-s − 0.401·67-s + 1.47·71-s − 0.123·73-s + 3.40·77-s − 0.113·79-s + 1.67·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.561936487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.561936487\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 - 36 T + p^{3} T^{2} \) |
| 11 | \( 1 - 64 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 59 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 160 T + p^{3} T^{2} \) |
| 29 | \( 1 + 57 T + p^{3} T^{2} \) |
| 31 | \( 1 - 164 T + p^{3} T^{2} \) |
| 37 | \( 1 + 321 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 8 T + p^{3} T^{2} \) |
| 47 | \( 1 - 84 T + p^{3} T^{2} \) |
| 53 | \( 1 - 478 T + p^{3} T^{2} \) |
| 59 | \( 1 + 32 T + p^{3} T^{2} \) |
| 61 | \( 1 - 415 T + p^{3} T^{2} \) |
| 67 | \( 1 + 220 T + p^{3} T^{2} \) |
| 71 | \( 1 - 884 T + p^{3} T^{2} \) |
| 73 | \( 1 + 77 T + p^{3} T^{2} \) |
| 79 | \( 1 + 80 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1268 T + p^{3} T^{2} \) |
| 89 | \( 1 - 123 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1346 T + p^{3} T^{2} \) |
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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
−10.21581134578005652593582290507, −9.156940287562783712994546429085, −8.386346567034038865660533441092, −7.54269803631698291244652272390, −6.82004439364284340113836912358, −5.34972537917882433732828392346, −4.65669181996994359650324409657, −3.69005499729225104266642198133, −2.06246584481550770345195590560, −1.01934273720162479571911548895,
1.01934273720162479571911548895, 2.06246584481550770345195590560, 3.69005499729225104266642198133, 4.65669181996994359650324409657, 5.34972537917882433732828392346, 6.82004439364284340113836912358, 7.54269803631698291244652272390, 8.386346567034038865660533441092, 9.156940287562783712994546429085, 10.21581134578005652593582290507
