| L(s) = 1 | + 10·9-s + 40·13-s − 12·25-s + 40·37-s − 140·49-s + 360·61-s − 120·73-s + 19·81-s + 40·97-s − 280·109-s + 400·117-s + 84·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 324·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 10/9·9-s + 3.07·13-s − 0.479·25-s + 1.08·37-s − 2.85·49-s + 5.90·61-s − 1.64·73-s + 0.234·81-s + 0.412·97-s − 2.56·109-s + 3.41·117-s + 0.694·121-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 + 1.91·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.788278145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.788278145\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 10 T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 10 p T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 770 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 282 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1222 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3138 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 310 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4290 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4218 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2838 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8950 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9282 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 11782 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13130 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10242 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
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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
−9.974561427876292750896703519481, −9.768474429410309681289591939578, −9.663648939869413507361184784566, −9.191041510116183709142512511829, −8.947483781057464908053361423468, −8.416854473485379466230653560019, −8.338575705543091190947850073912, −8.070420619120411324357258588727, −7.968327221503939783166704706975, −7.15750133427024150690561301329, −6.97629746554048834352943675060, −6.88612390360825807641377865161, −6.26404146557412019274676740834, −6.03493727108382524964382299464, −5.93954494763706842737278195764, −5.18149764702616660346862318318, −5.14774175202980619395963123481, −4.36018741684057147323911890198, −4.14808209795628261790621118818, −3.65614099408804659554077245026, −3.57961077157900907920586020120, −2.86565861104158780795544511738, −2.11908932813620542016928336159, −1.46302201217077532603767677200, −0.965923133741797720824814046479,
0.965923133741797720824814046479, 1.46302201217077532603767677200, 2.11908932813620542016928336159, 2.86565861104158780795544511738, 3.57961077157900907920586020120, 3.65614099408804659554077245026, 4.14808209795628261790621118818, 4.36018741684057147323911890198, 5.14774175202980619395963123481, 5.18149764702616660346862318318, 5.93954494763706842737278195764, 6.03493727108382524964382299464, 6.26404146557412019274676740834, 6.88612390360825807641377865161, 6.97629746554048834352943675060, 7.15750133427024150690561301329, 7.968327221503939783166704706975, 8.070420619120411324357258588727, 8.338575705543091190947850073912, 8.416854473485379466230653560019, 8.947483781057464908053361423468, 9.191041510116183709142512511829, 9.663648939869413507361184784566, 9.768474429410309681289591939578, 9.974561427876292750896703519481
