| L(s) = 1 | − 3-s + 2·5-s + 9-s − 4·13-s − 2·15-s + 17-s + 4·19-s + 4·23-s − 25-s − 27-s − 2·31-s + 4·39-s − 6·41-s + 12·43-s + 2·45-s + 10·47-s − 7·49-s − 51-s + 4·53-s − 4·57-s + 4·59-s − 2·61-s − 8·65-s − 12·67-s − 4·69-s − 8·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.10·13-s − 0.516·15-s + 0.242·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.359·31-s + 0.640·39-s − 0.937·41-s + 1.82·43-s + 0.298·45-s + 1.45·47-s − 49-s − 0.140·51-s + 0.549·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.992·65-s − 1.46·67-s − 0.481·69-s − 0.949·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.447285800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.447285800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.29133051807703, −12.09633443604582, −11.63401106578308, −10.95541566140851, −10.70551753466401, −10.18260626444792, −9.694817823826958, −9.400054755160354, −9.056919525506061, −8.343830597923590, −7.747117230800136, −7.335363789080990, −6.939868796525876, −6.477473727601516, −5.707017851524116, −5.568153809686275, −5.187833359534909, −4.461710055214019, −4.162737093462714, −3.253234185959760, −2.879800133779964, −2.218038652013777, −1.725240180493385, −1.046311471583356, −0.4523607593083030,
0.4523607593083030, 1.046311471583356, 1.725240180493385, 2.218038652013777, 2.879800133779964, 3.253234185959760, 4.162737093462714, 4.461710055214019, 5.187833359534909, 5.568153809686275, 5.707017851524116, 6.477473727601516, 6.939868796525876, 7.335363789080990, 7.747117230800136, 8.343830597923590, 9.056919525506061, 9.400054755160354, 9.694817823826958, 10.18260626444792, 10.70551753466401, 10.95541566140851, 11.63401106578308, 12.09633443604582, 12.29133051807703
