| L(s) = 1 | − 6·9-s − 8·25-s − 32·37-s + 27·81-s − 40·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 48·225-s + 227-s + ⋯ |
| L(s) = 1 | − 2·9-s − 8/5·25-s − 5.26·37-s + 3·81-s − 3.83·109-s + 3.63·121-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 − 4·169-s + 0.0760·173-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 + 0.0669·223-s + 16/5·225-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1553570629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1553570629\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 172 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−6.29706260960831750613270686979, −6.29291272895179281096972381301, −5.80898808755071121168295168030, −5.75108511108952738878256909680, −5.45339467614424216689196611988, −5.41961881032805411772378256416, −5.31621290832368697493696847915, −4.94892227586108203402590538798, −4.70131636481516264745009897332, −4.61877063343230696467379275239, −4.23988108097831991687896551062, −3.85710675981420994398508145602, −3.71782093766464687673275235877, −3.57551004297520296104969769845, −3.27696543770850333444925522531, −3.21533409493816358890282660618, −2.91716605522177245047374548203, −2.45038794539872617361860811970, −2.23624399118433364632000541698, −2.22475343223106269201023993982, −1.70430002482312423843975471442, −1.52049323604811346340513537758, −1.18272912914309086878057315245, −0.48232553811209032471025234907, −0.096655376359695728580090449323,
0.096655376359695728580090449323, 0.48232553811209032471025234907, 1.18272912914309086878057315245, 1.52049323604811346340513537758, 1.70430002482312423843975471442, 2.22475343223106269201023993982, 2.23624399118433364632000541698, 2.45038794539872617361860811970, 2.91716605522177245047374548203, 3.21533409493816358890282660618, 3.27696543770850333444925522531, 3.57551004297520296104969769845, 3.71782093766464687673275235877, 3.85710675981420994398508145602, 4.23988108097831991687896551062, 4.61877063343230696467379275239, 4.70131636481516264745009897332, 4.94892227586108203402590538798, 5.31621290832368697493696847915, 5.41961881032805411772378256416, 5.45339467614424216689196611988, 5.75108511108952738878256909680, 5.80898808755071121168295168030, 6.29291272895179281096972381301, 6.29706260960831750613270686979
