| L(s) = 1 | + 4·2-s + 10·4-s + 10·7-s + 20·8-s − 2·9-s − 6·13-s + 40·14-s + 35·16-s − 8·18-s − 24·26-s + 100·28-s + 4·29-s + 56·32-s − 20·36-s + 10·37-s + 16·47-s + 43·49-s − 60·52-s + 200·56-s + 16·58-s + 22·61-s − 20·63-s + 84·64-s − 40·72-s + 12·73-s + 40·74-s + 34·79-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 3.77·7-s + 7.07·8-s − 2/3·9-s − 1.66·13-s + 10.6·14-s + 35/4·16-s − 1.88·18-s − 4.70·26-s + 18.8·28-s + 0.742·29-s + 9.89·32-s − 3.33·36-s + 1.64·37-s + 2.33·47-s + 43/7·49-s − 8.32·52-s + 26.7·56-s + 2.10·58-s + 2.81·61-s − 2.51·63-s + 21/2·64-s − 4.71·72-s + 1.40·73-s + 4.64·74-s + 3.82·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(67.96596010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(67.96596010\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 35 T^{2} + 544 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 9 T^{2} + 560 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 5 T^{2} + 384 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 91 T^{2} + 3784 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 143 T^{2} + 8368 T^{4} - 143 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 766 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 8 T + 93 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 81 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 115 T^{2} + 9312 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 11 T + 114 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 117 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 133 T^{2} + 14296 T^{4} + 133 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 320 T^{2} + 41374 T^{4} - 320 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.60808963845234723784652321178, −6.16301545003711982548648108591, −5.92739318247823280687298326774, −5.69032602622908653510423076280, −5.38522889979832200756657608483, −5.31611625451242825294091496093, −5.24407233698133812719026404565, −5.06829470931469848649815236220, −4.99598123622721561171684245662, −4.50514807680803187153381606090, −4.32978101891680160418753567586, −4.25449104993498039730157823362, −4.21435301830112040252756670200, −3.86354345729159526014236897649, −3.61211647204185011524979466040, −3.11793430939152688161468129420, −2.94617678224004847199214692661, −2.64100028273112984720574543324, −2.41509112563544248808065595068, −2.19384758200483174757776531505, −2.15164465023259595211167077661, −1.63433480163886803347015552337, −1.51095203082177497676708257270, −0.924285400919668761143694720428, −0.76595900528964349667836063529,
0.76595900528964349667836063529, 0.924285400919668761143694720428, 1.51095203082177497676708257270, 1.63433480163886803347015552337, 2.15164465023259595211167077661, 2.19384758200483174757776531505, 2.41509112563544248808065595068, 2.64100028273112984720574543324, 2.94617678224004847199214692661, 3.11793430939152688161468129420, 3.61211647204185011524979466040, 3.86354345729159526014236897649, 4.21435301830112040252756670200, 4.25449104993498039730157823362, 4.32978101891680160418753567586, 4.50514807680803187153381606090, 4.99598123622721561171684245662, 5.06829470931469848649815236220, 5.24407233698133812719026404565, 5.31611625451242825294091496093, 5.38522889979832200756657608483, 5.69032602622908653510423076280, 5.92739318247823280687298326774, 6.16301545003711982548648108591, 6.60808963845234723784652321178
