| L(s) = 1 | − 2·4-s − 4·7-s + 4·13-s − 12·19-s + 8·28-s − 12·31-s + 32·43-s + 10·49-s − 8·52-s − 12·61-s + 32·67-s + 8·73-s + 24·76-s − 16·91-s − 12·97-s + 8·103-s − 20·109-s − 8·121-s + 24·124-s + 127-s + 131-s + 48·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s + 1.10·13-s − 2.75·19-s + 1.51·28-s − 2.15·31-s + 4.87·43-s + 10/7·49-s − 1.10·52-s − 1.53·61-s + 3.90·67-s + 0.936·73-s + 2.75·76-s − 1.67·91-s − 1.21·97-s + 0.788·103-s − 1.91·109-s − 0.727·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.750903989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.750903989\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 8 T^{2} - 62 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 38 T^{2} + 814 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T^{2} - 206 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 102 T^{2} + 4238 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 134 T^{2} + 7726 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 16 T + 145 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 72 T^{2} + 2834 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 7474 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 72 T^{2} + 5378 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 16 T + 193 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 228 T^{2} + 22358 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 153 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 156 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 6 T + 198 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.92255482980075318688327747176, −5.61281283990504193364249315068, −5.50519270641639253691677408282, −5.43028382550787418478381249904, −5.19252728717438051103655802593, −4.64289572440631920742306516850, −4.56372839373375156049358739435, −4.44390563320090702716707713269, −4.31531397440309977131504800723, −3.95226716380957776637140885948, −3.93314929493145864733254626263, −3.79982383215032584855469498119, −3.44949875741231211218771234282, −3.39269368204857476957178879043, −2.96665099948059103351335290400, −2.77930032839905193205662642684, −2.58896670155777034773437290572, −2.20089372016998826278509058374, −2.17495111496905461270505807055, −1.77784557471176990639422065423, −1.67421762134005123252944882847, −1.14405194727934654029345811422, −0.55484243526708857145693600189, −0.50812155415059351283782559206, −0.49984678476585637580904345489,
0.49984678476585637580904345489, 0.50812155415059351283782559206, 0.55484243526708857145693600189, 1.14405194727934654029345811422, 1.67421762134005123252944882847, 1.77784557471176990639422065423, 2.17495111496905461270505807055, 2.20089372016998826278509058374, 2.58896670155777034773437290572, 2.77930032839905193205662642684, 2.96665099948059103351335290400, 3.39269368204857476957178879043, 3.44949875741231211218771234282, 3.79982383215032584855469498119, 3.93314929493145864733254626263, 3.95226716380957776637140885948, 4.31531397440309977131504800723, 4.44390563320090702716707713269, 4.56372839373375156049358739435, 4.64289572440631920742306516850, 5.19252728717438051103655802593, 5.43028382550787418478381249904, 5.50519270641639253691677408282, 5.61281283990504193364249315068, 5.92255482980075318688327747176
