| L(s) = 1 | − 3-s + 2·7-s + 9-s + 11-s + 4·13-s − 6·17-s + 4·19-s − 2·21-s + 6·23-s − 5·25-s − 27-s − 6·29-s + 8·31-s − 33-s + 10·37-s − 4·39-s + 6·41-s − 8·43-s − 6·47-s − 3·49-s + 6·51-s − 4·57-s − 8·61-s + 2·63-s + 4·67-s − 6·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 1.45·17-s + 0.917·19-s − 0.436·21-s + 1.25·23-s − 25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s + 1.64·37-s − 0.640·39-s + 0.937·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s + 0.840·51-s − 0.529·57-s − 1.02·61-s + 0.251·63-s + 0.488·67-s − 0.722·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.690886324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.690886324\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.150127955345467646802362904166, −8.272692985838996706791775704327, −7.58826295647403442525037964215, −6.58952139537910579113504192674, −6.04212512673208031413734655017, −5.01949978616669493961505925185, −4.38779849097927031276804988726, −3.36588825019911542049550749608, −2.00808309976256142330449222770, −0.928729343829663921711511365168,
0.928729343829663921711511365168, 2.00808309976256142330449222770, 3.36588825019911542049550749608, 4.38779849097927031276804988726, 5.01949978616669493961505925185, 6.04212512673208031413734655017, 6.58952139537910579113504192674, 7.58826295647403442525037964215, 8.272692985838996706791775704327, 9.150127955345467646802362904166
