| L(s) = 1 | − 12·5-s − 27·9-s + 172·13-s + 852·17-s − 1.14e3·25-s − 2.36e3·29-s + 860·37-s + 4.50e3·41-s + 324·45-s + 914·49-s + 3.20e3·53-s − 4.22e3·61-s − 2.06e3·65-s + 8.13e3·73-s + 729·81-s − 1.02e4·85-s − 4.09e3·89-s − 5.88e3·97-s + 2.33e4·101-s − 3.50e4·109-s + 2.40e4·113-s − 4.64e3·117-s − 5.71e3·121-s + 2.16e4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.479·5-s − 1/3·9-s + 1.01·13-s + 2.94·17-s − 1.82·25-s − 2.81·29-s + 0.628·37-s + 2.67·41-s + 4/25·45-s + 0.380·49-s + 1.14·53-s − 1.13·61-s − 0.488·65-s + 1.52·73-s + 1/9·81-s − 1.41·85-s − 0.516·89-s − 0.625·97-s + 2.29·101-s − 2.95·109-s + 1.88·113-s − 0.339·117-s − 0.390·121-s + 1.38·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.256079341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.256079341\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 6 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 914 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5710 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 86 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 426 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 256754 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 190 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 1182 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 582958 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 430 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2250 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 190 p^{2} T^{2} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9619394 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 1602 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 11602610 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2114 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 38587634 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 41865410 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4066 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 47103314 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10900850 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2046 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2942 T + p^{4} T^{2} )^{2} \) |
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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
−12.03988071947171974455410682871, −11.69999369681912621665587196364, −10.99839995521059931415841775055, −10.91426823494330350015846874727, −10.05983995032866721134367769504, −9.452945944375264307741247581544, −9.407196776574690830858750515600, −8.480996090906597778643611105520, −7.80808189777775110640886760262, −7.71901308396404731537621710080, −7.23076540704023610713578176636, −6.08063625989352125452673947908, −5.62886533152600222385891439528, −5.62176299669672406973026601249, −4.34595465206099892134215743540, −3.63621772811811975782906290114, −3.48608396289622738793019560478, −2.35189459784699262891500270857, −1.38604910845612020864876934199, −0.58540537027730307109301864504,
0.58540537027730307109301864504, 1.38604910845612020864876934199, 2.35189459784699262891500270857, 3.48608396289622738793019560478, 3.63621772811811975782906290114, 4.34595465206099892134215743540, 5.62176299669672406973026601249, 5.62886533152600222385891439528, 6.08063625989352125452673947908, 7.23076540704023610713578176636, 7.71901308396404731537621710080, 7.80808189777775110640886760262, 8.480996090906597778643611105520, 9.407196776574690830858750515600, 9.452945944375264307741247581544, 10.05983995032866721134367769504, 10.91426823494330350015846874727, 10.99839995521059931415841775055, 11.69999369681912621665587196364, 12.03988071947171974455410682871
