| L(s) = 1 | − 9-s + 4·11-s + 10·23-s − 2·29-s − 10·37-s − 26·43-s + 4·53-s + 8·67-s + 30·71-s + 2·79-s + 9·81-s − 4·99-s − 38·107-s − 2·109-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.20·11-s + 2.08·23-s − 0.371·29-s − 1.64·37-s − 3.96·43-s + 0.549·53-s + 0.977·67-s + 3.56·71-s + 0.225·79-s + 81-s − 0.402·99-s − 3.67·107-s − 0.191·109-s − 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.930925197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.930925197\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 29 T^{2} + 552 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 5 T^{2} + 492 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 5 T + 26 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + T + 32 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 80 T^{2} + 3102 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 5 T + 54 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 65 T^{2} + 2292 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 144 T^{2} + 9182 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 17 T^{2} - 5676 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - T + 132 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 177 T^{2} + 15704 T^{4} + 177 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 201 T^{2} + 20036 T^{4} + 201 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 104 T^{2} + 17742 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.46316053778971824177840189587, −5.08434881389621974488476948317, −5.08268978469961938004537153574, −4.87829158053605709234198019784, −4.78277617961525594534316757392, −4.47848552412884312035981604202, −4.28813604698553195848793229041, −4.02078568031897357282223350843, −3.69337322645432867222923269437, −3.69276483311509616652090336454, −3.56848770577539234829551759025, −3.49953599433493285207344770429, −3.19187988723790917264305152506, −2.80338951104058831952165929529, −2.69766351015186077194525684041, −2.62442319105276240362486412642, −2.39563822009889538797079567213, −1.89233884989080443203199131605, −1.69972328288393644215439918222, −1.62596957182085604122361225356, −1.49475659547766379652651811782, −1.18297934367126775877501121785, −0.63765460048183692330330707970, −0.56544740279164568365905722591, −0.36656218033564945048951429922,
0.36656218033564945048951429922, 0.56544740279164568365905722591, 0.63765460048183692330330707970, 1.18297934367126775877501121785, 1.49475659547766379652651811782, 1.62596957182085604122361225356, 1.69972328288393644215439918222, 1.89233884989080443203199131605, 2.39563822009889538797079567213, 2.62442319105276240362486412642, 2.69766351015186077194525684041, 2.80338951104058831952165929529, 3.19187988723790917264305152506, 3.49953599433493285207344770429, 3.56848770577539234829551759025, 3.69276483311509616652090336454, 3.69337322645432867222923269437, 4.02078568031897357282223350843, 4.28813604698553195848793229041, 4.47848552412884312035981604202, 4.78277617961525594534316757392, 4.87829158053605709234198019784, 5.08268978469961938004537153574, 5.08434881389621974488476948317, 5.46316053778971824177840189587
