| L(s) = 1 | − 7-s − 3·9-s + 4·13-s − 6·19-s − 23-s + 6·29-s + 2·31-s + 6·37-s + 6·41-s − 8·43-s − 2·47-s + 49-s − 6·53-s + 8·59-s − 10·61-s + 3·63-s + 8·67-s − 8·71-s + 10·73-s + 8·79-s + 9·81-s + 6·83-s + 8·89-s − 4·91-s − 4·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.10·13-s − 1.37·19-s − 0.208·23-s + 1.11·29-s + 0.359·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.291·47-s + 1/7·49-s − 0.824·53-s + 1.04·59-s − 1.28·61-s + 0.377·63-s + 0.977·67-s − 0.949·71-s + 1.17·73-s + 0.900·79-s + 81-s + 0.658·83-s + 0.847·89-s − 0.419·91-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.16022839506470, −12.65841554878697, −11.99296345385268, −11.78502354203672, −11.11585952358129, −10.77058301681136, −10.45000767895042, −9.759843074564171, −9.319236190045377, −8.801764986615049, −8.376197951497527, −8.060706798362604, −7.550413345690341, −6.597183815901118, −6.366723252384919, −6.183465098797262, −5.411600210169819, −4.928915987511123, −4.297595879225510, −3.790447440539035, −3.276866398191040, −2.649652411027400, −2.246455672840499, −1.410387008768227, −0.7305061542031630, 0,
0.7305061542031630, 1.410387008768227, 2.246455672840499, 2.649652411027400, 3.276866398191040, 3.790447440539035, 4.297595879225510, 4.928915987511123, 5.411600210169819, 6.183465098797262, 6.366723252384919, 6.597183815901118, 7.550413345690341, 8.060706798362604, 8.376197951497527, 8.801764986615049, 9.319236190045377, 9.759843074564171, 10.45000767895042, 10.77058301681136, 11.11585952358129, 11.78502354203672, 11.99296345385268, 12.65841554878697, 13.16022839506470
