| L(s) = 1 | + 4·3-s + 3·4-s + 12·7-s + 7·9-s + 12·12-s − 32·13-s − 7·16-s − 4·19-s + 48·21-s − 8·27-s + 36·28-s − 36·31-s + 21·36-s + 32·37-s − 128·39-s − 32·43-s − 28·48-s + 10·49-s − 96·52-s − 16·57-s + 164·61-s + 84·63-s − 69·64-s − 48·67-s + 148·73-s − 12·76-s + 276·79-s + ⋯ |
| L(s) = 1 | + 4/3·3-s + 3/4·4-s + 12/7·7-s + 7/9·9-s + 12-s − 2.46·13-s − 0.437·16-s − 0.210·19-s + 16/7·21-s − 0.296·27-s + 9/7·28-s − 1.16·31-s + 7/12·36-s + 0.864·37-s − 3.28·39-s − 0.744·43-s − 0.583·48-s + 0.204·49-s − 1.84·52-s − 0.280·57-s + 2.68·61-s + 4/3·63-s − 1.07·64-s − 0.716·67-s + 2.02·73-s − 0.157·76-s + 3.49·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.626050638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.626050638\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 4 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 222 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 558 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )( 1 + 44 T + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 702 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 558 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1998 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5598 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6942 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 5598 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 138 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4958 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 166 T + p^{2} T^{2} )^{2} \) |
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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
−14.72925744034027549497102066514, −14.17589883281362921748119557958, −13.75326723590149048788464616642, −12.92948695490250188622702395879, −12.42429911978288214491320735219, −11.71616684749326208355240428995, −11.35440298561855003841836180509, −10.79912101692157862322654685651, −9.907638259880481509168622235017, −9.533394723911941401829027941286, −8.770001270166904773054877015488, −8.167733565251724439666970532232, −7.52840572078571648059290912058, −7.37743318295281245481235974274, −6.41867482361789393790001923895, −5.05884099906590924491252519753, −4.83701565982152218221761493790, −3.62403313672418599504592721933, −2.29695851358570863524361320685, −2.13418391705532504473109597688,
2.13418391705532504473109597688, 2.29695851358570863524361320685, 3.62403313672418599504592721933, 4.83701565982152218221761493790, 5.05884099906590924491252519753, 6.41867482361789393790001923895, 7.37743318295281245481235974274, 7.52840572078571648059290912058, 8.167733565251724439666970532232, 8.770001270166904773054877015488, 9.533394723911941401829027941286, 9.907638259880481509168622235017, 10.79912101692157862322654685651, 11.35440298561855003841836180509, 11.71616684749326208355240428995, 12.42429911978288214491320735219, 12.92948695490250188622702395879, 13.75326723590149048788464616642, 14.17589883281362921748119557958, 14.72925744034027549497102066514
