| L(s) = 1 | + 84·29-s − 36·41-s − 98·49-s − 44·61-s − 156·89-s + 396·101-s + 364·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 238·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 2.89·29-s − 0.878·41-s − 2·49-s − 0.721·61-s − 1.75·89-s + 3.92·101-s + 3.33·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.995214473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.995214473\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )( 1 + 56 T + p^{2} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )( 1 + 96 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 78 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )( 1 + 144 T + p^{2} T^{2} ) \) |
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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
−8.423146071225194891445536908567, −8.329840280434429595240268912809, −7.896868988922363850144708121319, −7.43741344705362038238349644144, −7.04595657436222383984941229370, −6.75899460754924532921479462085, −6.17678763101371326451872437831, −6.16116840980861175754445882903, −5.65743154460599204297325950437, −4.88619085309983097019005788199, −4.74559464802729090551860975010, −4.65021694096081774540435020373, −3.86510646697353275612459208867, −3.43346892070780205789256485335, −3.00903053671178171695048744654, −2.71185527923394299123981032292, −1.97983280651345409395405237256, −1.61624006166066034727976135805, −0.902899393372356573996191420601, −0.44477499229467265226900397263,
0.44477499229467265226900397263, 0.902899393372356573996191420601, 1.61624006166066034727976135805, 1.97983280651345409395405237256, 2.71185527923394299123981032292, 3.00903053671178171695048744654, 3.43346892070780205789256485335, 3.86510646697353275612459208867, 4.65021694096081774540435020373, 4.74559464802729090551860975010, 4.88619085309983097019005788199, 5.65743154460599204297325950437, 6.16116840980861175754445882903, 6.17678763101371326451872437831, 6.75899460754924532921479462085, 7.04595657436222383984941229370, 7.43741344705362038238349644144, 7.896868988922363850144708121319, 8.329840280434429595240268912809, 8.423146071225194891445536908567
