| L(s) = 1 | − 8·7-s + 14·9-s + 48·23-s − 116·41-s − 232·47-s − 156·49-s − 112·63-s − 15·81-s − 260·89-s + 456·103-s − 462·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 384·161-s + 163-s + 167-s − 588·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 8/7·7-s + 14/9·9-s + 2.08·23-s − 2.82·41-s − 4.93·47-s − 3.18·49-s − 1.77·63-s − 0.185·81-s − 2.92·89-s + 4.42·103-s − 3.81·121-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 − 2.38·161-s + 0.00613·163-s + 0.00598·167-s − 3.47·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.117143378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.117143378\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2}( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 21 p T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 294 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 457 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 183 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 474 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1346 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 582 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 29 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 2114 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 1218 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2562 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 3878 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 7119 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8926 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 9433 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 382 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13239 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 65 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 862 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.42479374375996469370030267225, −6.39999488123246369628207950302, −6.25588028190792368628674066373, −6.13198333444477723544728978830, −5.54909964401449460085246564822, −5.24587341905396388367128740116, −5.19777832921206083586543525533, −4.87537762859115155936997581271, −4.76561693434438966763212902168, −4.59933133088526150279127859376, −4.58522720893278999089246217688, −3.74121674928260205739050679319, −3.70772783100156799552205112644, −3.64346379635240805369756708359, −3.41720292831007668912931372424, −3.07997699704079668492046083908, −2.89620551755097259199236635350, −2.48248132212202134235110943547, −2.44699249096413701966265039783, −1.56574714822080356315668017333, −1.54297359502670391902639247490, −1.39379003736127946557603955629, −1.36609802716641609400760634761, −0.35882660863397326003695879967, −0.22025067412084512235709927434,
0.22025067412084512235709927434, 0.35882660863397326003695879967, 1.36609802716641609400760634761, 1.39379003736127946557603955629, 1.54297359502670391902639247490, 1.56574714822080356315668017333, 2.44699249096413701966265039783, 2.48248132212202134235110943547, 2.89620551755097259199236635350, 3.07997699704079668492046083908, 3.41720292831007668912931372424, 3.64346379635240805369756708359, 3.70772783100156799552205112644, 3.74121674928260205739050679319, 4.58522720893278999089246217688, 4.59933133088526150279127859376, 4.76561693434438966763212902168, 4.87537762859115155936997581271, 5.19777832921206083586543525533, 5.24587341905396388367128740116, 5.54909964401449460085246564822, 6.13198333444477723544728978830, 6.25588028190792368628674066373, 6.39999488123246369628207950302, 6.42479374375996469370030267225
