| L(s) = 1 | − 3·2-s + 4·4-s + 5·7-s − 3·8-s − 3·9-s + 6·11-s − 15·14-s + 3·16-s − 6·17-s + 9·18-s − 12·19-s − 18·22-s + 3·23-s + 20·28-s − 6·31-s − 6·32-s + 18·34-s − 12·36-s + 4·37-s + 36·38-s + 6·41-s − 2·43-s + 24·44-s − 9·46-s + 18·49-s − 15·56-s − 9·61-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s + 1.88·7-s − 1.06·8-s − 9-s + 1.80·11-s − 4.00·14-s + 3/4·16-s − 1.45·17-s + 2.12·18-s − 2.75·19-s − 3.83·22-s + 0.625·23-s + 3.77·28-s − 1.07·31-s − 1.06·32-s + 3.08·34-s − 2·36-s + 0.657·37-s + 5.83·38-s + 0.937·41-s − 0.304·43-s + 3.61·44-s − 1.32·46-s + 18/7·49-s − 2.00·56-s − 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4803738100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4803738100\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
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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
−10.90804375699040920097146219675, −10.72488767366802184456444912170, −10.27271836141236963311890415781, −9.328328095508950100191348210440, −9.181332958396641308130262522563, −8.810968646450715539072660421692, −8.676847126394447138055225033601, −8.046579033712958228665655895707, −7.975471045820858145695861668465, −7.07826082841924579101631793550, −6.84651061662732352503594710611, −6.02477940787102558058016703567, −5.89620916215843162358149701948, −4.83966341408845863366616686658, −4.34381341930807007258902733836, −4.04685548189866241618608465926, −2.87272187373347892593797480937, −1.85645975410394283444172087028, −1.77679179106632974928780606415, −0.59211553365999991701221482196,
0.59211553365999991701221482196, 1.77679179106632974928780606415, 1.85645975410394283444172087028, 2.87272187373347892593797480937, 4.04685548189866241618608465926, 4.34381341930807007258902733836, 4.83966341408845863366616686658, 5.89620916215843162358149701948, 6.02477940787102558058016703567, 6.84651061662732352503594710611, 7.07826082841924579101631793550, 7.975471045820858145695861668465, 8.046579033712958228665655895707, 8.676847126394447138055225033601, 8.810968646450715539072660421692, 9.181332958396641308130262522563, 9.328328095508950100191348210440, 10.27271836141236963311890415781, 10.72488767366802184456444912170, 10.90804375699040920097146219675
