| L(s) = 1 | + 5-s + 11-s + 2·13-s − 4·17-s + 6·23-s + 25-s + 4·29-s + 4·31-s − 8·37-s − 6·41-s − 4·43-s − 8·47-s + 4·53-s + 55-s + 2·59-s + 10·61-s + 2·65-s + 8·67-s − 8·71-s + 4·73-s + 10·79-s + 2·83-s − 4·85-s + 16·89-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s + 0.554·13-s − 0.970·17-s + 1.25·23-s + 1/5·25-s + 0.742·29-s + 0.718·31-s − 1.31·37-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 0.549·53-s + 0.134·55-s + 0.260·59-s + 1.28·61-s + 0.248·65-s + 0.977·67-s − 0.949·71-s + 0.468·73-s + 1.12·79-s + 0.219·83-s − 0.433·85-s + 1.69·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.187709950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.187709950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−12.41487412809911, −12.00860442888793, −11.50971986168025, −11.15331331818279, −10.59340015391221, −10.28426113045189, −9.745553594365829, −9.286309450355120, −8.742362811965013, −8.492985862485537, −8.069391374831893, −7.269228841163665, −6.797807126816701, −6.547620950669647, −6.141066607041846, −5.253123850734174, −5.125043285200045, −4.554815760346410, −3.901525434606497, −3.388851828531735, −2.920093805249705, −2.219933430577243, −1.758391501741803, −1.091573146140387, −0.4897163199431848,
0.4897163199431848, 1.091573146140387, 1.758391501741803, 2.219933430577243, 2.920093805249705, 3.388851828531735, 3.901525434606497, 4.554815760346410, 5.125043285200045, 5.253123850734174, 6.141066607041846, 6.547620950669647, 6.797807126816701, 7.269228841163665, 8.069391374831893, 8.492985862485537, 8.742362811965013, 9.286309450355120, 9.745553594365829, 10.28426113045189, 10.59340015391221, 11.15331331818279, 11.50971986168025, 12.00860442888793, 12.41487412809911
