| L(s) = 1 | + 72·13-s + 64·25-s + 144·37-s − 52·49-s − 144·61-s + 224·73-s + 416·97-s + 504·109-s + 452·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.56e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 5.53·13-s + 2.55·25-s + 3.89·37-s − 1.06·49-s − 2.36·61-s + 3.06·73-s + 4.28·97-s + 4.62·109-s + 3.73·121-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 + 15.1·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(22.57020112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.57020112\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 226 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 434 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 238 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1520 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1850 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 36 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 2480 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 910 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3122 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 880 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 562 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 36 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 5134 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 1582 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 11834 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 8002 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 7904 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{4} \) |
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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
−6.24581960492475403996825225703, −6.19710354915341018854906907111, −6.00199216645760551865128171124, −5.65487741962150828302186550809, −5.52510147718540070394885442421, −4.99271892044536017125735480346, −4.95273144512609675010603710781, −4.77797160382348273723587864842, −4.43498183139070041143050492129, −4.30114877837366467979853018582, −4.02368623284356445703828905157, −3.81162537174320868931913517640, −3.56969884054493449255976056327, −3.23715309029776660775566690829, −3.21265809862949163570068451340, −3.18843273468208098088748479450, −2.71787949717832760495332015342, −2.31818496450662347507345108403, −2.06695141705515783611716983226, −1.62177220926973201313887640709, −1.55837696782949090172155977308, −1.09303320892179352384823201806, −0.831931732237830513893955171838, −0.76741683850571098636273368182, −0.61089215933263526475090730111,
0.61089215933263526475090730111, 0.76741683850571098636273368182, 0.831931732237830513893955171838, 1.09303320892179352384823201806, 1.55837696782949090172155977308, 1.62177220926973201313887640709, 2.06695141705515783611716983226, 2.31818496450662347507345108403, 2.71787949717832760495332015342, 3.18843273468208098088748479450, 3.21265809862949163570068451340, 3.23715309029776660775566690829, 3.56969884054493449255976056327, 3.81162537174320868931913517640, 4.02368623284356445703828905157, 4.30114877837366467979853018582, 4.43498183139070041143050492129, 4.77797160382348273723587864842, 4.95273144512609675010603710781, 4.99271892044536017125735480346, 5.52510147718540070394885442421, 5.65487741962150828302186550809, 6.00199216645760551865128171124, 6.19710354915341018854906907111, 6.24581960492475403996825225703
