| L(s) = 1 | − 2·7-s + 2·13-s + 2·19-s + 2·31-s + 2·43-s + 49-s + 2·61-s − 2·67-s − 4·91-s + 2·97-s − 4·103-s − 2·109-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·7-s + 2·13-s + 2·19-s + 2·31-s + 2·43-s + 49-s + 2·61-s − 2·67-s − 4·91-s + 2·97-s − 4·103-s − 2·109-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.123231962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123231962\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.511175960404781618649421398952, −9.341590092977878629828543916595, −9.054463240882008076704081073943, −8.488175431717945062313005350818, −8.102257043226013423307724383503, −7.77444311268645063685427607118, −7.17833273073807590623599083198, −6.79555112695010867992420096465, −6.47279737514731729846813250396, −6.12224649955494156101339490121, −5.67741807178035927669926686561, −5.47907828137923836687614048949, −4.67081895796964342886463253640, −4.19746307343099414941982481620, −3.54627576406584498228687204239, −3.50745555893537077493993703579, −2.79957114848715855936262809137, −2.60133102228583573717590057644, −1.35574886675700468075506851066, −0.927879736999085279198485373471,
0.927879736999085279198485373471, 1.35574886675700468075506851066, 2.60133102228583573717590057644, 2.79957114848715855936262809137, 3.50745555893537077493993703579, 3.54627576406584498228687204239, 4.19746307343099414941982481620, 4.67081895796964342886463253640, 5.47907828137923836687614048949, 5.67741807178035927669926686561, 6.12224649955494156101339490121, 6.47279737514731729846813250396, 6.79555112695010867992420096465, 7.17833273073807590623599083198, 7.77444311268645063685427607118, 8.102257043226013423307724383503, 8.488175431717945062313005350818, 9.054463240882008076704081073943, 9.341590092977878629828543916595, 9.511175960404781618649421398952
