| L(s) = 1 | − 12·5-s + 40·13-s + 16·17-s − 142·25-s + 92·29-s − 328·37-s + 624·41-s − 238·49-s − 532·53-s − 264·61-s − 480·65-s + 492·73-s − 192·85-s + 2.78e3·89-s − 604·97-s − 2.93e3·101-s − 3.12e3·109-s − 3.10e3·113-s − 870·121-s + 3.63e3·125-s + 127-s + 131-s + 137-s + 139-s − 1.10e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.07·5-s + 0.853·13-s + 0.228·17-s − 1.13·25-s + 0.589·29-s − 1.45·37-s + 2.37·41-s − 0.693·49-s − 1.37·53-s − 0.554·61-s − 0.915·65-s + 0.788·73-s − 0.245·85-s + 3.31·89-s − 0.632·97-s − 2.88·101-s − 2.74·109-s − 2.58·113-s − 0.653·121-s + 2.60·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.632·145-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 34 p T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 870 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6550 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4338 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 59134 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 164 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 312 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 20186 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 178974 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 346246 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 132 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 343478 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 257070 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 931870 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 195606 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1392 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 302 T + p^{3} T^{2} )^{2} \) |
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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
−8.168167927255731955086518353937, −8.151983544079724472110008609149, −7.77092133738538295394211340733, −7.48925166552715238872542985600, −6.84013548926934587914514500904, −6.62444761569389701250182031264, −6.05368044855299303696892849446, −5.85081188139248230707909492016, −5.20150023306796514361114272350, −4.92522210563609866280458026059, −4.17708255379714852694468200092, −4.09953707129669894010031102254, −3.57012330100525943800995470349, −3.23067451718472970834016554478, −2.60282025855601100967287450490, −2.09107140383675182861094831733, −1.35637473763195318488072520121, −1.01339538556350284911064830202, 0, 0,
1.01339538556350284911064830202, 1.35637473763195318488072520121, 2.09107140383675182861094831733, 2.60282025855601100967287450490, 3.23067451718472970834016554478, 3.57012330100525943800995470349, 4.09953707129669894010031102254, 4.17708255379714852694468200092, 4.92522210563609866280458026059, 5.20150023306796514361114272350, 5.85081188139248230707909492016, 6.05368044855299303696892849446, 6.62444761569389701250182031264, 6.84013548926934587914514500904, 7.48925166552715238872542985600, 7.77092133738538295394211340733, 8.151983544079724472110008609149, 8.168167927255731955086518353937
