| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s − 13-s − 15-s + 6·17-s − 2·19-s + 2·21-s + 6·23-s + 25-s − 27-s + 9·29-s − 2·31-s − 2·35-s + 37-s + 39-s + 43-s + 45-s − 9·47-s − 3·49-s − 6·51-s + 3·53-s + 2·57-s + 3·59-s − 10·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 1.45·17-s − 0.458·19-s + 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s − 0.359·31-s − 0.338·35-s + 0.164·37-s + 0.160·39-s + 0.152·43-s + 0.149·45-s − 1.31·47-s − 3/7·49-s − 0.840·51-s + 0.412·53-s + 0.264·57-s + 0.390·59-s − 1.28·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.636824593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.636824593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−7.68380992651346876417869189806, −6.82587365252976698463339462867, −6.46570991976190070250157052670, −5.67547309251349386778087237855, −5.10022011156649028216530397294, −4.37171192342118209638450332281, −3.30838734942701179162889763048, −2.79830567124980585525967595777, −1.58305105566969559002741659059, −0.66345573026568874362169927990,
0.66345573026568874362169927990, 1.58305105566969559002741659059, 2.79830567124980585525967595777, 3.30838734942701179162889763048, 4.37171192342118209638450332281, 5.10022011156649028216530397294, 5.67547309251349386778087237855, 6.46570991976190070250157052670, 6.82587365252976698463339462867, 7.68380992651346876417869189806
