L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 11-s + 3·13-s − 14-s + 16-s − 4·17-s − 6·19-s + 22-s + 3·23-s + 3·26-s − 28-s − 6·29-s + 2·31-s + 32-s − 4·34-s − 7·37-s − 6·38-s + 2·41-s − 2·43-s + 44-s + 3·46-s − 7·47-s + 49-s + 3·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.301·11-s + 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.37·19-s + 0.213·22-s + 0.625·23-s + 0.588·26-s − 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s − 0.685·34-s − 1.15·37-s − 0.973·38-s + 0.312·41-s − 0.304·43-s + 0.150·44-s + 0.442·46-s − 1.02·47-s + 1/7·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.11359084769488475321407278889, −6.53662113777136479146258096105, −6.10220475862277263393048289514, −5.27013112189652878765325725401, −4.47894670751705835740178215862, −3.88257487856613638280494739832, −3.18756374841505916237765341239, −2.26085513507293130329855634861, −1.45727076908437670636132754331, 0,
1.45727076908437670636132754331, 2.26085513507293130329855634861, 3.18756374841505916237765341239, 3.88257487856613638280494739832, 4.47894670751705835740178215862, 5.27013112189652878765325725401, 6.10220475862277263393048289514, 6.53662113777136479146258096105, 7.11359084769488475321407278889