| L(s) = 1 | − 18·5-s − 7·9-s − 32·13-s + 14·17-s + 193·25-s − 64·29-s − 2·37-s + 80·41-s + 126·45-s + 14·53-s − 158·61-s + 576·65-s − 286·73-s − 32·81-s − 252·85-s + 194·89-s + 176·97-s + 30·101-s + 226·109-s + 224·117-s + 233·121-s − 1.56e3·125-s + 127-s + 131-s + 137-s + 139-s + 1.15e3·145-s + ⋯ |
| L(s) = 1 | − 3.59·5-s − 7/9·9-s − 2.46·13-s + 0.823·17-s + 7.71·25-s − 2.20·29-s − 0.0540·37-s + 1.95·41-s + 14/5·45-s + 0.264·53-s − 2.59·61-s + 8.86·65-s − 3.91·73-s − 0.395·81-s − 2.96·85-s + 2.17·89-s + 1.81·97-s + 0.297·101-s + 2.07·109-s + 1.91·117-s + 1.92·121-s − 12.5·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 7.94·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4527268007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4527268007\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 7 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 9 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 233 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 601 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 697 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1801 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2098 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2807 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4153 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 79 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8857 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 7778 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 143 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11257 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13714 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 88 T + p^{2} T^{2} )^{2} \) |
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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
−9.241469677393549660983173810617, −8.806515657471662291110289769834, −8.770879447275554489682412083692, −7.939236293930860062248801817273, −7.76507993730426716718865338769, −7.47860959110570895089628262215, −7.23070544006016712593315010439, −7.20020540283511436664335860566, −6.10069978439213187211998301415, −5.83094642829043535980258202896, −5.04224514233906133356805699394, −4.71275087149820965822857292171, −4.44660887192312401649971201084, −3.93467931047523397666025701646, −3.50719770100004282061531214357, −2.97770790151589403408354670265, −2.80858471337558816284755212766, −1.82051250533610879457080468340, −0.56026313298801034541009365446, −0.35591139938768181442901152755,
0.35591139938768181442901152755, 0.56026313298801034541009365446, 1.82051250533610879457080468340, 2.80858471337558816284755212766, 2.97770790151589403408354670265, 3.50719770100004282061531214357, 3.93467931047523397666025701646, 4.44660887192312401649971201084, 4.71275087149820965822857292171, 5.04224514233906133356805699394, 5.83094642829043535980258202896, 6.10069978439213187211998301415, 7.20020540283511436664335860566, 7.23070544006016712593315010439, 7.47860959110570895089628262215, 7.76507993730426716718865338769, 7.939236293930860062248801817273, 8.770879447275554489682412083692, 8.806515657471662291110289769834, 9.241469677393549660983173810617
