L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·11-s − 2·17-s + 19-s + 4·21-s + 2·23-s − 5·25-s − 27-s + 6·29-s − 6·31-s − 4·33-s + 8·37-s + 10·41-s − 12·43-s − 10·47-s + 9·49-s + 2·51-s − 2·53-s − 57-s + 4·59-s + 10·61-s − 4·63-s − 2·69-s + 16·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.485·17-s + 0.229·19-s + 0.872·21-s + 0.417·23-s − 25-s − 0.192·27-s + 1.11·29-s − 1.07·31-s − 0.696·33-s + 1.31·37-s + 1.56·41-s − 1.82·43-s − 1.45·47-s + 9/7·49-s + 0.280·51-s − 0.274·53-s − 0.132·57-s + 0.520·59-s + 1.28·61-s − 0.503·63-s − 0.240·69-s + 1.89·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.202111681048451602860500584503, −7.14484973381679933288499588457, −6.56793180927898505204276009439, −6.15303436289493204195564662977, −5.24974769470397520513045103465, −4.18996617241742471056324636487, −3.58834969751350496793839399629, −2.59096801605995758443809230131, −1.25485200713650100256263649356, 0,
1.25485200713650100256263649356, 2.59096801605995758443809230131, 3.58834969751350496793839399629, 4.18996617241742471056324636487, 5.24974769470397520513045103465, 6.15303436289493204195564662977, 6.56793180927898505204276009439, 7.14484973381679933288499588457, 8.202111681048451602860500584503