| L(s) = 1 | + 44·5-s − 92·13-s + 384·17-s + 968·25-s + 412·29-s − 300·37-s − 68·49-s + 1.30e3·53-s − 156·61-s − 4.04e3·65-s + 1.26e3·81-s + 1.68e4·85-s + 1.85e3·97-s + 3.79e3·101-s + 2.33e3·109-s − 4.47e3·113-s + 1.61e4·125-s + 127-s + 131-s + 137-s + 139-s + 1.81e4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.93·5-s − 1.96·13-s + 5.47·17-s + 7.74·25-s + 2.63·29-s − 1.33·37-s − 0.198·49-s + 3.36·53-s − 0.327·61-s − 7.72·65-s + 1.73·81-s + 21.5·85-s + 1.94·97-s + 3.73·101-s + 2.04·109-s − 3.72·113-s + 11.5·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 10.3·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(27.34669705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.34669705\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 1262 T^{4} + p^{12} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 3050318 T^{4} + p^{12} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 46 T + 1058 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 89434798 T^{4} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 10814 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 206 T + 21218 T^{2} - 206 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 20862 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 150 T + 11250 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 40498 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 11798591182 T^{4} + p^{12} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 199646 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 650 T + 211250 T^{2} - 650 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 878 p^{4} T^{4} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 78 T + 3042 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 123064007662 T^{4} + p^{12} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 581342 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 517010 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 298648058542 T^{4} + p^{12} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 578162 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 464 T + p^{3} T^{2} )^{4} \) |
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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
−7.57771093479830170796445556351, −7.08905169218016160761473533387, −6.83255636956535778830630576236, −6.76777146624414154957104501040, −6.25735751552326424173955829352, −6.11404855110593517759497803336, −5.95120518406224164900918631574, −5.57806732391055630203862994972, −5.47938608427385595840416276788, −5.32221601844909085213590057383, −5.19472193075313141018444449403, −4.84305889544423433269792926740, −4.65607823787197854691941357074, −4.16152320454934550461872291377, −3.52742088288099054857736747522, −3.27806816098684393484094731038, −3.24727252525557145520415739928, −2.73750596983686504386918714068, −2.49245756238321598431887803574, −2.23197568879232002540408932477, −1.79266415347160940507018781201, −1.69249694803514167599263037332, −1.04692725358005885663544287182, −0.890680600191145394233636333162, −0.70967894729695865639604611529,
0.70967894729695865639604611529, 0.890680600191145394233636333162, 1.04692725358005885663544287182, 1.69249694803514167599263037332, 1.79266415347160940507018781201, 2.23197568879232002540408932477, 2.49245756238321598431887803574, 2.73750596983686504386918714068, 3.24727252525557145520415739928, 3.27806816098684393484094731038, 3.52742088288099054857736747522, 4.16152320454934550461872291377, 4.65607823787197854691941357074, 4.84305889544423433269792926740, 5.19472193075313141018444449403, 5.32221601844909085213590057383, 5.47938608427385595840416276788, 5.57806732391055630203862994972, 5.95120518406224164900918631574, 6.11404855110593517759497803336, 6.25735751552326424173955829352, 6.76777146624414154957104501040, 6.83255636956535778830630576236, 7.08905169218016160761473533387, 7.57771093479830170796445556351
