| L(s) = 1 | + 10·3-s + 27·9-s + 54·11-s − 212·19-s + 250·25-s − 190·27-s + 540·33-s − 1.56e3·41-s − 290·43-s − 686·49-s − 2.12e3·57-s + 2.53e3·59-s + 70·67-s + 860·73-s + 2.50e3·75-s − 1.90e3·81-s + 1.91e3·97-s + 1.45e3·99-s + 289·121-s − 1.56e4·123-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 9-s + 1.48·11-s − 2.55·19-s + 2·25-s − 1.35·27-s + 2.84·33-s − 5.96·41-s − 1.02·43-s − 2·49-s − 4.92·57-s + 5.60·59-s + 0.127·67-s + 1.37·73-s + 3.84·75-s − 2.60·81-s + 1.99·97-s + 1.48·99-s + 0.217·121-s − 11.4·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.871508315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.871508315\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 10 T + 73 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2}( 1 - 18 T - 1007 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 90 T + 3187 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} )( 1 + 90 T + 3187 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 106 T + 4377 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 522 T + p^{3} T^{2} )^{2}( 1 + 522 T + 203563 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 290 T + p^{3} T^{2} )^{2}( 1 - 290 T + 4593 T^{2} - 290 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 846 T + p^{3} T^{2} )^{2}( 1 - 846 T + 510337 T^{2} - 846 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2}( 1 + 70 T - 295863 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 430 T - 204117 T^{2} - 430 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1350 T + 1250713 T^{2} - 1350 p^{3} T^{3} + p^{6} T^{4} )( 1 + 1350 T + 1250713 T^{2} + 1350 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - 1026 T + p^{3} T^{2} )^{2}( 1 + 1026 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 1910 T + p^{3} T^{2} )^{2}( 1 + 1910 T + 2735427 T^{2} + 1910 p^{3} T^{3} + p^{6} T^{4} ) \) |
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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
−8.311787890658717220965073147597, −8.129964840347520799365912689873, −7.979034235973103118471150593724, −7.14923827240832357575423272964, −6.99903264525272264846370709356, −6.85867036140484470746083094839, −6.63302502360949433173849856117, −6.63179594014455852462308513129, −6.14714566189480955240267632380, −5.69073415036995966759942902340, −5.44119209353046166768965351589, −5.01471384360770097642929638257, −4.73737694015616700887430739704, −4.64912258619858455838136558438, −4.06505723430491364987538101950, −3.64063756572069425656410915328, −3.59696099809913025315015012158, −3.28566091072765673831111391334, −3.12643567217658837640037646422, −2.37530420536968059725168081287, −2.20334974744964915249823449000, −1.91809523621287953045682967845, −1.57236919810019882761827874756, −0.937042301878864689528427944014, −0.23509117610912112358516562633,
0.23509117610912112358516562633, 0.937042301878864689528427944014, 1.57236919810019882761827874756, 1.91809523621287953045682967845, 2.20334974744964915249823449000, 2.37530420536968059725168081287, 3.12643567217658837640037646422, 3.28566091072765673831111391334, 3.59696099809913025315015012158, 3.64063756572069425656410915328, 4.06505723430491364987538101950, 4.64912258619858455838136558438, 4.73737694015616700887430739704, 5.01471384360770097642929638257, 5.44119209353046166768965351589, 5.69073415036995966759942902340, 6.14714566189480955240267632380, 6.63179594014455852462308513129, 6.63302502360949433173849856117, 6.85867036140484470746083094839, 6.99903264525272264846370709356, 7.14923827240832357575423272964, 7.979034235973103118471150593724, 8.129964840347520799365912689873, 8.311787890658717220965073147597
