L(s) = 1 | − 3-s − 5·7-s − 2·9-s + 6·13-s − 4·17-s + 2·19-s + 5·21-s + 5·27-s − 10·29-s − 2·31-s − 4·37-s − 6·39-s − 3·41-s − 3·43-s + 5·47-s + 18·49-s + 4·51-s + 6·53-s − 2·57-s − 10·59-s + 5·61-s + 10·63-s − 7·67-s − 2·71-s + 4·73-s + 8·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.88·7-s − 2/3·9-s + 1.66·13-s − 0.970·17-s + 0.458·19-s + 1.09·21-s + 0.962·27-s − 1.85·29-s − 0.359·31-s − 0.657·37-s − 0.960·39-s − 0.468·41-s − 0.457·43-s + 0.729·47-s + 18/7·49-s + 0.560·51-s + 0.824·53-s − 0.264·57-s − 1.30·59-s + 0.640·61-s + 1.25·63-s − 0.855·67-s − 0.237·71-s + 0.468·73-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 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
−15.08955481110445, −14.07679763209972, −13.67186275868326, −13.21371772265378, −12.90589053281166, −12.23860521204738, −11.69929823446555, −11.21661930994499, −10.60380676087976, −10.39343342707358, −9.396995485997984, −9.177992596346068, −8.717589796241257, −8.003610456370809, −7.162462910281341, −6.695850726669822, −6.248442031764451, −5.719961445424118, −5.395942277191626, −4.355710244267195, −3.591663103313913, −3.407216159753289, −2.601815587730933, −1.731887007728217, −0.6803272746417076, 0,
0.6803272746417076, 1.731887007728217, 2.601815587730933, 3.407216159753289, 3.591663103313913, 4.355710244267195, 5.395942277191626, 5.719961445424118, 6.248442031764451, 6.695850726669822, 7.162462910281341, 8.003610456370809, 8.717589796241257, 9.177992596346068, 9.396995485997984, 10.39343342707358, 10.60380676087976, 11.21661930994499, 11.69929823446555, 12.23860521204738, 12.90589053281166, 13.21371772265378, 13.67186275868326, 14.07679763209972, 15.08955481110445