| L(s) = 1 | + 360·17-s + 284·25-s + 216·41-s − 1.34e3·49-s + 4.42e3·73-s − 3.67e3·89-s + 760·97-s + 8.85e3·113-s + 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.64e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 5.13·17-s + 2.27·25-s + 0.822·41-s − 3.93·49-s + 7.09·73-s − 4.37·89-s + 0.795·97-s + 7.37·113-s + 4·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.20·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.07967037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.07967037\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 674 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 1322 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 262 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 13534 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 18722 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 31246 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 100106 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 158614 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 51694 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 59182 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 305782 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 123290 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 588070 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 497666 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1106 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 963890 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 183530 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 918 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 190 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.38065961444394514039507099289, −7.12042485671404348946149375329, −6.81413368224547355817572429102, −6.55090821679825224629709907910, −6.45509157921449141693658915182, −5.95997791176097202630506000045, −5.80286742499403989703370860618, −5.65647732287199770790469440265, −5.26304131498748058550176080354, −5.21441709402519607279620709709, −4.81558821065441690727371782061, −4.60653679284354441168629692777, −4.53003537850584610330309770400, −3.73966307152885802703518787955, −3.54770112797699938034778810813, −3.37784826713696623167923017599, −3.28854477800799987510054958763, −2.99269839248672319317280663115, −2.62693423749089587531718313518, −2.07432196772251983217338255004, −1.79587838159385328714875736922, −1.32586054326697453404228349761, −0.903649556444670541909337484573, −0.888161018309458046769633287055, −0.48208755968438760267178356362,
0.48208755968438760267178356362, 0.888161018309458046769633287055, 0.903649556444670541909337484573, 1.32586054326697453404228349761, 1.79587838159385328714875736922, 2.07432196772251983217338255004, 2.62693423749089587531718313518, 2.99269839248672319317280663115, 3.28854477800799987510054958763, 3.37784826713696623167923017599, 3.54770112797699938034778810813, 3.73966307152885802703518787955, 4.53003537850584610330309770400, 4.60653679284354441168629692777, 4.81558821065441690727371782061, 5.21441709402519607279620709709, 5.26304131498748058550176080354, 5.65647732287199770790469440265, 5.80286742499403989703370860618, 5.95997791176097202630506000045, 6.45509157921449141693658915182, 6.55090821679825224629709907910, 6.81413368224547355817572429102, 7.12042485671404348946149375329, 7.38065961444394514039507099289
