| L(s) = 1 | − 2·4-s − 5·9-s + 14·11-s − 5·16-s + 16·19-s − 4·29-s − 2·31-s + 10·36-s + 4·41-s − 28·44-s − 20·59-s + 24·61-s + 20·64-s + 4·71-s − 32·76-s + 34·79-s + 9·81-s + 20·89-s − 70·99-s − 36·101-s − 30·109-s + 8·116-s + 95·121-s + 4·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s − 5/3·9-s + 4.22·11-s − 5/4·16-s + 3.67·19-s − 0.742·29-s − 0.359·31-s + 5/3·36-s + 0.624·41-s − 4.22·44-s − 2.60·59-s + 3.07·61-s + 5/2·64-s + 0.474·71-s − 3.67·76-s + 3.82·79-s + 81-s + 2.11·89-s − 7.03·99-s − 3.58·101-s − 2.87·109-s + 0.742·116-s + 8.63·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.459360318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.459360318\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 33 T^{2} + 536 T^{4} + 33 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 846 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 950 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 21 T^{2} + 1952 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4$ | \( 1 + 37 T^{2} + 2904 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 193 T^{2} + 14856 T^{4} + 193 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 + 192 T^{2} + 17006 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 2 T + 110 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 17 T + 222 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4$ | \( 1 - 44 T^{2} + 5814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 170 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.35145469917342474814226061172, −7.22084336413811525146972161696, −7.07069623654531515843659565612, −6.68155554192467102717080903714, −6.44427976919527264043786368559, −6.43392109823624342081227793492, −6.16863735485524568032576467136, −5.84726771721344651112631139962, −5.53070809099941376982567569217, −5.20888133934643334661959641509, −5.10191221180345160267606468974, −5.00486608862277657645388655703, −4.59959358451985462742552877344, −4.05840635262312553796066659356, −3.96555899274983907686224242528, −3.78773641512783063365353027357, −3.68798326217590414048863110178, −3.25503110037288496639848261214, −3.02214980313415212572016383877, −2.51384454552368115694380085033, −2.31329215939164738681910428802, −1.65432359396682364961687111457, −1.32139496593749583994855245692, −1.03980805065612102299840916477, −0.51784198567911311843301615293,
0.51784198567911311843301615293, 1.03980805065612102299840916477, 1.32139496593749583994855245692, 1.65432359396682364961687111457, 2.31329215939164738681910428802, 2.51384454552368115694380085033, 3.02214980313415212572016383877, 3.25503110037288496639848261214, 3.68798326217590414048863110178, 3.78773641512783063365353027357, 3.96555899274983907686224242528, 4.05840635262312553796066659356, 4.59959358451985462742552877344, 5.00486608862277657645388655703, 5.10191221180345160267606468974, 5.20888133934643334661959641509, 5.53070809099941376982567569217, 5.84726771721344651112631139962, 6.16863735485524568032576467136, 6.43392109823624342081227793492, 6.44427976919527264043786368559, 6.68155554192467102717080903714, 7.07069623654531515843659565612, 7.22084336413811525146972161696, 7.35145469917342474814226061172
