| L(s) = 1 | − 8·7-s − 2·16-s − 20·25-s + 8·37-s − 16·43-s + 32·49-s + 32·67-s + 16·112-s − 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 160·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 1/2·16-s − 4·25-s + 1.31·37-s − 2.43·43-s + 32/7·49-s + 3.90·67-s + 1.51·112-s − 1.45·121-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 + 0.0760·173-s + 12.0·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06258971829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06258971829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + p T^{2} )^{4} \) |
| 7 | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 + T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 - 2 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 706 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 5582 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 5218 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 2098 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 11458 T^{4} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.35455342629090684011593786563, −5.22595127753690979181185297964, −5.14896710525280000379208156444, −4.73723693706320594158189786040, −4.43580422544033213529847670309, −4.41941528038749870043094090800, −4.18778371573229499211207188745, −4.15472121072797075555539370555, −4.06556031681943537257684683186, −3.78214199514477582146663832276, −3.54971607626692794962323264546, −3.49372945752301837296835472332, −3.31814760939105171016652567270, −3.25514935920689086068508982094, −3.20292970072892527012841240749, −2.68925340447673800383751780212, −2.57644769854430966341711731838, −2.37662434067493117404871385785, −2.28504892322483222189447952248, −2.10947101182841777143617389416, −1.71249064078519069153370415083, −1.61457414064169123493372154367, −1.09494271072425852550321804776, −0.67044928474312393554136277868, −0.083050644597596274201820538595,
0.083050644597596274201820538595, 0.67044928474312393554136277868, 1.09494271072425852550321804776, 1.61457414064169123493372154367, 1.71249064078519069153370415083, 2.10947101182841777143617389416, 2.28504892322483222189447952248, 2.37662434067493117404871385785, 2.57644769854430966341711731838, 2.68925340447673800383751780212, 3.20292970072892527012841240749, 3.25514935920689086068508982094, 3.31814760939105171016652567270, 3.49372945752301837296835472332, 3.54971607626692794962323264546, 3.78214199514477582146663832276, 4.06556031681943537257684683186, 4.15472121072797075555539370555, 4.18778371573229499211207188745, 4.41941528038749870043094090800, 4.43580422544033213529847670309, 4.73723693706320594158189786040, 5.14896710525280000379208156444, 5.22595127753690979181185297964, 5.35455342629090684011593786563
Plot not available for L-functions of degree greater than 10.
