L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 13-s − 15-s − 2·17-s − 4·19-s − 4·21-s + 8·23-s + 25-s − 27-s − 2·29-s − 8·31-s + 4·35-s − 2·37-s − 39-s − 6·41-s − 12·43-s + 45-s + 9·49-s + 2·51-s − 10·53-s + 4·57-s + 10·61-s + 4·63-s + 65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.872·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.676·35-s − 0.328·37-s − 0.160·39-s − 0.937·41-s − 1.82·43-s + 0.149·45-s + 9/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s + 1.28·61-s + 0.503·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.82676073814800, −16.12368470843964, −15.34824609825697, −14.77984469801321, −14.59955697035254, −13.71347914839238, −13.10416777557712, −12.78279041541939, −11.85248972642698, −11.36110555767048, −10.93310309311402, −10.50137196286168, −9.707572418469722, −8.875556090809650, −8.544464322370439, −7.795898843888803, −6.978996210136638, −6.613747486694067, −5.605719489211694, −5.180358394209827, −4.630030192694407, −3.868942667235526, −2.843751344603721, −1.758116976427070, −1.436564479698511, 0,
1.436564479698511, 1.758116976427070, 2.843751344603721, 3.868942667235526, 4.630030192694407, 5.180358394209827, 5.605719489211694, 6.613747486694067, 6.978996210136638, 7.795898843888803, 8.544464322370439, 8.875556090809650, 9.707572418469722, 10.50137196286168, 10.93310309311402, 11.36110555767048, 11.85248972642698, 12.78279041541939, 13.10416777557712, 13.71347914839238, 14.59955697035254, 14.77984469801321, 15.34824609825697, 16.12368470843964, 16.82676073814800