| L(s) = 1 | + 3-s + 7-s + 9-s − 6·11-s + 13-s − 3·17-s + 4·19-s + 21-s − 3·23-s + 27-s + 3·29-s − 5·31-s − 6·33-s + 10·37-s + 39-s + 9·41-s − 43-s + 49-s − 3·51-s − 9·53-s + 4·57-s − 9·59-s + 11·61-s + 63-s − 4·67-s − 3·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s − 0.727·17-s + 0.917·19-s + 0.218·21-s − 0.625·23-s + 0.192·27-s + 0.557·29-s − 0.898·31-s − 1.04·33-s + 1.64·37-s + 0.160·39-s + 1.40·41-s − 0.152·43-s + 1/7·49-s − 0.420·51-s − 1.23·53-s + 0.529·57-s − 1.17·59-s + 1.40·61-s + 0.125·63-s − 0.488·67-s − 0.361·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.259540709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.259540709\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.908220868583684923536465850228, −7.36953421110109550625254581421, −6.42336021955059773313831189363, −5.65305633217775258429084621644, −4.96747280757645114313885628023, −4.31135753195991685499576206191, −3.36818272481441082106306892461, −2.60268529220797634734675891967, −1.98721572396478945544870655698, −0.69814422051997673964396588563,
0.69814422051997673964396588563, 1.98721572396478945544870655698, 2.60268529220797634734675891967, 3.36818272481441082106306892461, 4.31135753195991685499576206191, 4.96747280757645114313885628023, 5.65305633217775258429084621644, 6.42336021955059773313831189363, 7.36953421110109550625254581421, 7.908220868583684923536465850228
