L(s) = 1 | − 3-s + 2·5-s + 9-s + 6·13-s − 2·15-s + 2·17-s + 4·19-s − 4·23-s − 25-s − 27-s + 10·29-s + 8·31-s − 6·37-s − 6·39-s + 2·41-s + 4·43-s + 2·45-s − 8·47-s − 2·51-s + 10·53-s − 4·57-s + 12·59-s − 2·61-s + 12·65-s − 12·67-s + 4·69-s − 12·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.66·13-s − 0.516·15-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 1.43·31-s − 0.986·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s − 0.280·51-s + 1.37·53-s − 0.529·57-s + 1.56·59-s − 0.256·61-s + 1.48·65-s − 1.46·67-s + 0.481·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.561925011\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.561925011\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.72877858431690597029846138467, −6.76421550604128576334431406016, −6.24237084218937118492748887699, −5.76832301033763836485883884275, −5.11573786125903126012137049497, −4.25264275449370137452074838908, −3.46692171542106744111654218383, −2.58276427230269739881283371701, −1.51080472195466306773023973270, −0.877088698922972818064521338359,
0.877088698922972818064521338359, 1.51080472195466306773023973270, 2.58276427230269739881283371701, 3.46692171542106744111654218383, 4.25264275449370137452074838908, 5.11573786125903126012137049497, 5.76832301033763836485883884275, 6.24237084218937118492748887699, 6.76421550604128576334431406016, 7.72877858431690597029846138467