| L(s) = 1 | + 2-s + 2·3-s − 4-s + 5·5-s + 2·6-s − 3·8-s + 3·9-s + 5·10-s − 2·12-s + 10·15-s − 16-s − 3·17-s + 3·18-s − 19-s − 5·20-s − 6·24-s + 11·25-s + 4·27-s + 10·30-s + 7·31-s + 5·32-s − 3·34-s − 3·36-s − 38-s − 15·40-s + 15·45-s − 2·48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 2.23·5-s + 0.816·6-s − 1.06·8-s + 9-s + 1.58·10-s − 0.577·12-s + 2.58·15-s − 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.229·19-s − 1.11·20-s − 1.22·24-s + 11/5·25-s + 0.769·27-s + 1.82·30-s + 1.25·31-s + 0.883·32-s − 0.514·34-s − 1/2·36-s − 0.162·38-s − 2.37·40-s + 2.23·45-s − 0.288·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.040501053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.040501053\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 165 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
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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
−8.301276033902248886072544324981, −7.83767424717188548791787990887, −7.08539267885969964936271339550, −6.63184230370292439861888314224, −6.33789817811069174496896442851, −5.83086745494965605328944737782, −5.47317821006990102026865378357, −4.85591809358781683871283314618, −4.59548332478518888234100145371, −3.90451504588036497167163213578, −3.39968332980185008642977401955, −2.75886969069619919972935688817, −2.27508195481470203047502626025, −1.94786254432634666588022972622, −0.987798022315836519928665919699,
0.987798022315836519928665919699, 1.94786254432634666588022972622, 2.27508195481470203047502626025, 2.75886969069619919972935688817, 3.39968332980185008642977401955, 3.90451504588036497167163213578, 4.59548332478518888234100145371, 4.85591809358781683871283314618, 5.47317821006990102026865378357, 5.83086745494965605328944737782, 6.33789817811069174496896442851, 6.63184230370292439861888314224, 7.08539267885969964936271339550, 7.83767424717188548791787990887, 8.301276033902248886072544324981
