L(s) = 1 | − 2-s + 4-s − 5-s + 3·7-s − 8-s + 10-s − 2·11-s − 6·13-s − 3·14-s + 16-s + 2·17-s − 5·19-s − 20-s + 2·22-s − 4·23-s − 4·25-s + 6·26-s + 3·28-s + 5·29-s + 2·31-s − 32-s − 2·34-s − 3·35-s + 8·37-s + 5·38-s + 40-s − 7·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 0.353·8-s + 0.316·10-s − 0.603·11-s − 1.66·13-s − 0.801·14-s + 1/4·16-s + 0.485·17-s − 1.14·19-s − 0.223·20-s + 0.426·22-s − 0.834·23-s − 4/5·25-s + 1.17·26-s + 0.566·28-s + 0.928·29-s + 0.359·31-s − 0.176·32-s − 0.342·34-s − 0.507·35-s + 1.31·37-s + 0.811·38-s + 0.158·40-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1062 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1062 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.617683989518697456184327254574, −8.366079000518683854365832031484, −7.983926917574178165017009870672, −7.31857923616660990948628249613, −6.20827475938246748031936324003, −5.05154692225783055374248162365, −4.32947828426301183770623455344, −2.78464504521222655185101352564, −1.78766840056115711664367411136, 0,
1.78766840056115711664367411136, 2.78464504521222655185101352564, 4.32947828426301183770623455344, 5.05154692225783055374248162365, 6.20827475938246748031936324003, 7.31857923616660990948628249613, 7.983926917574178165017009870672, 8.366079000518683854365832031484, 9.617683989518697456184327254574