| L(s) = 1 | − 2·3-s − 7-s + 9-s + 4·13-s + 6·17-s + 2·19-s + 2·21-s − 5·25-s + 4·27-s + 6·29-s + 4·31-s − 2·37-s − 8·39-s + 6·41-s + 8·43-s + 12·47-s + 49-s − 12·51-s − 6·53-s − 4·57-s − 6·59-s − 8·61-s − 63-s − 4·67-s + 2·73-s + 10·75-s − 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s + 0.458·19-s + 0.436·21-s − 25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s − 1.28·39-s + 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 1.68·51-s − 0.824·53-s − 0.529·57-s − 0.781·59-s − 1.02·61-s − 0.125·63-s − 0.488·67-s + 0.234·73-s + 1.15·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9372638439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9372638439\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.08163105431410064794151744331, −10.39623059414136659793705857031, −9.479912708108790860766004869191, −8.328602349022309226048251750698, −7.29200576171944643329126204500, −6.04992362572003306544038558924, −5.73625972355331222443538857924, −4.39126934619586232082133319537, −3.09628357761679869111062402047, −1.00403210582709168291350506092,
1.00403210582709168291350506092, 3.09628357761679869111062402047, 4.39126934619586232082133319537, 5.73625972355331222443538857924, 6.04992362572003306544038558924, 7.29200576171944643329126204500, 8.328602349022309226048251750698, 9.479912708108790860766004869191, 10.39623059414136659793705857031, 11.08163105431410064794151744331
