| L(s) = 1 | + 3-s − 4·7-s − 2·9-s + 13-s + 3·17-s + 2·19-s − 4·21-s − 3·23-s − 5·25-s − 5·27-s − 3·29-s − 4·31-s + 2·37-s + 39-s + 12·41-s − 43-s + 6·47-s + 9·49-s + 3·51-s + 3·53-s + 2·57-s − 12·59-s + 11·61-s + 8·63-s − 4·67-s − 3·69-s + 6·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s − 2/3·9-s + 0.277·13-s + 0.727·17-s + 0.458·19-s − 0.872·21-s − 0.625·23-s − 25-s − 0.962·27-s − 0.557·29-s − 0.718·31-s + 0.328·37-s + 0.160·39-s + 1.87·41-s − 0.152·43-s + 0.875·47-s + 9/7·49-s + 0.420·51-s + 0.412·53-s + 0.264·57-s − 1.56·59-s + 1.40·61-s + 1.00·63-s − 0.488·67-s − 0.361·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484956023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484956023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.895792581516946305808222079763, −7.54259637648536086348115946262, −6.55729087168729982070996148010, −5.88039660736249968130313285044, −5.47521822027838667545426259539, −4.06864793232484688687177028428, −3.55165405239133717957288259189, −2.87919152319618945471954628181, −2.05809752079212717232359853699, −0.58704505136065432234983693841,
0.58704505136065432234983693841, 2.05809752079212717232359853699, 2.87919152319618945471954628181, 3.55165405239133717957288259189, 4.06864793232484688687177028428, 5.47521822027838667545426259539, 5.88039660736249968130313285044, 6.55729087168729982070996148010, 7.54259637648536086348115946262, 7.895792581516946305808222079763
