| L(s) = 1 | − 2·3-s + 9·9-s − 14·11-s − 4·17-s − 68·19-s − 50·25-s − 46·27-s + 28·33-s − 46·41-s − 14·43-s − 98·49-s + 8·51-s + 136·57-s + 82·59-s − 62·67-s + 284·73-s + 100·75-s + 92·81-s + 316·83-s + 584·89-s − 94·97-s − 126·99-s − 356·107-s − 196·113-s + 75·121-s + 92·123-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 9-s − 1.27·11-s − 0.235·17-s − 3.57·19-s − 2·25-s − 1.70·27-s + 0.848·33-s − 1.12·41-s − 0.325·43-s − 2·49-s + 8/51·51-s + 2.38·57-s + 1.38·59-s − 0.925·67-s + 3.89·73-s + 4/3·75-s + 1.13·81-s + 3.80·83-s + 6.56·89-s − 0.969·97-s − 1.27·99-s − 3.32·107-s − 1.73·113-s + 0.619·121-s + 0.747·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02878333567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02878333567\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 2 T - 5 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2}( 1 - 14 T + 75 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T - 285 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 34 T + 795 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2}( 1 - 46 T + 435 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2}( 1 - 14 T - 1653 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2}( 1 + 82 T + 3243 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{2}( 1 - 62 T - 645 T^{2} - 62 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T + 14835 T^{2} - 142 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 158 T + 18075 T^{2} - 158 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{2}( 1 - 94 T - 573 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} ) \) |
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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
−8.239250983184838986602391625123, −8.144226942867350179289784120522, −7.906285540939428678381516570805, −7.79821841375333178167974744753, −7.35559461648517605057575214501, −6.84889113580086924308438887044, −6.79284769949381463056990599331, −6.33986146257032069400196014639, −6.32360266227045506248923348098, −6.26571177013840456992249888660, −5.55191677030443730843363395626, −5.48916510687452433046839623771, −5.02682580451703300276257624243, −4.84715323855188248578779565684, −4.66490009082766624102209146143, −4.09884060198201843513334723586, −3.93949538069999100343654279335, −3.61765525968149976829530167657, −3.41707612399392380782285624012, −2.63446597125448341766197391574, −2.09005104688007122989052549880, −2.00337422327535804492861253018, −1.94206552493513016339430807832, −0.826279793173763450755958912974, −0.05090209206659717547647466417,
0.05090209206659717547647466417, 0.826279793173763450755958912974, 1.94206552493513016339430807832, 2.00337422327535804492861253018, 2.09005104688007122989052549880, 2.63446597125448341766197391574, 3.41707612399392380782285624012, 3.61765525968149976829530167657, 3.93949538069999100343654279335, 4.09884060198201843513334723586, 4.66490009082766624102209146143, 4.84715323855188248578779565684, 5.02682580451703300276257624243, 5.48916510687452433046839623771, 5.55191677030443730843363395626, 6.26571177013840456992249888660, 6.32360266227045506248923348098, 6.33986146257032069400196014639, 6.79284769949381463056990599331, 6.84889113580086924308438887044, 7.35559461648517605057575214501, 7.79821841375333178167974744753, 7.906285540939428678381516570805, 8.144226942867350179289784120522, 8.239250983184838986602391625123
