| L(s) = 1 | − 5-s − 2·7-s − 4·11-s − 13-s − 4·17-s − 6·19-s + 25-s − 4·29-s − 6·31-s + 2·35-s − 2·37-s + 10·41-s − 8·43-s − 3·49-s − 4·53-s + 4·55-s − 4·59-s + 2·61-s + 65-s − 6·67-s + 8·71-s + 10·73-s + 8·77-s + 4·79-s + 12·83-s + 4·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.20·11-s − 0.277·13-s − 0.970·17-s − 1.37·19-s + 1/5·25-s − 0.742·29-s − 1.07·31-s + 0.338·35-s − 0.328·37-s + 1.56·41-s − 1.21·43-s − 3/7·49-s − 0.549·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.124·65-s − 0.733·67-s + 0.949·71-s + 1.17·73-s + 0.911·77-s + 0.450·79-s + 1.31·83-s + 0.433·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3945835645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3945835645\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.82673926200934680021833611955, −6.93074553881435814437025789262, −6.48810154380045141774792433982, −5.64114309148133421460722185969, −4.92652199503071726349853066282, −4.18421111927993800217176182295, −3.45549319386058842150766602173, −2.60100981058052605492976778982, −1.91821589413904103629950614841, −0.27929181090522686834994922981,
0.27929181090522686834994922981, 1.91821589413904103629950614841, 2.60100981058052605492976778982, 3.45549319386058842150766602173, 4.18421111927993800217176182295, 4.92652199503071726349853066282, 5.64114309148133421460722185969, 6.48810154380045141774792433982, 6.93074553881435814437025789262, 7.82673926200934680021833611955
