L(s) = 1 | − 7-s − 3·9-s − 4·13-s − 4·17-s + 4·19-s − 8·23-s + 2·29-s − 8·31-s + 8·37-s + 6·41-s − 8·43-s − 8·47-s + 49-s − 4·59-s − 6·61-s + 3·63-s − 8·67-s + 12·71-s + 4·73-s − 4·79-s + 9·81-s − 10·89-s + 4·91-s + 12·97-s − 18·101-s + 8·103-s − 8·107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 9-s − 1.10·13-s − 0.970·17-s + 0.917·19-s − 1.66·23-s + 0.371·29-s − 1.43·31-s + 1.31·37-s + 0.937·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.520·59-s − 0.768·61-s + 0.377·63-s − 0.977·67-s + 1.42·71-s + 0.468·73-s − 0.450·79-s + 81-s − 1.05·89-s + 0.419·91-s + 1.21·97-s − 1.79·101-s + 0.788·103-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.856855657203209729848353217196, −9.303960674528025373762335683124, −8.257247054559279919607874840373, −7.45810191926908839910156976570, −6.39624533754705627341258933393, −5.55847041937131165989762890423, −4.50446761263334875285095208967, −3.24456810415544046511400683849, −2.18229815491186358015345347579, 0,
2.18229815491186358015345347579, 3.24456810415544046511400683849, 4.50446761263334875285095208967, 5.55847041937131165989762890423, 6.39624533754705627341258933393, 7.45810191926908839910156976570, 8.257247054559279919607874840373, 9.303960674528025373762335683124, 9.856855657203209729848353217196