| L(s) = 1 | − 8·7-s + 7·16-s + 24·25-s − 56·31-s + 36·49-s − 24·73-s + 8·79-s − 16·97-s − 24·103-s − 56·112-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s − 192·175-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 7/4·16-s + 24/5·25-s − 10.0·31-s + 36/7·49-s − 2.80·73-s + 0.900·79-s − 1.62·97-s − 2.36·103-s − 5.29·112-s − 0.727·121-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 + 3.07·169-s + 0.0760·173-s − 14.5·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2617303362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2617303362\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 7 T^{4} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T )^{8} \) |
| good | 5 | \( ( 1 - 12 T^{2} + 71 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} + 111 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + 378 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 14 T^{2} - 189 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 84 T^{2} + 2807 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 7 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 - 110 T^{2} + 5523 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 12 T^{2} + 2663 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 92 T^{2} + 4314 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 - 24 T^{2} + 3266 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 52 T^{2} - 522 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 20 T^{2} + 4218 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 236 T^{2} + 23871 T^{4} + 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 6 T + 140 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 120 T^{2} + 13538 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 204 T^{2} + 25511 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.88684416121596189171358957123, −3.85883202616619750356043760452, −3.67337520973268467818594404088, −3.58498268007406818316704038461, −3.54124683492880151760624993694, −3.31186567848027221997734856023, −3.25006493111005529755774307288, −3.23056591876677218344386182232, −3.08573186008268237773925384356, −2.88300001516468106532973696540, −2.74098675570014514149170913157, −2.61719772603265234114357990371, −2.56440454755023916427193009404, −2.43218289155926521795765659996, −1.99973945372021215876467136246, −1.93738935616731485086851826324, −1.75881668261124496464759877801, −1.72338282366061538859149420260, −1.50082046866480656820952361138, −1.31331939543842238212354345425, −1.14940378535561257656517445991, −0.867520158661744525773945397522, −0.55365197348122578372680248305, −0.37185068443447219278213697201, −0.085707919404388907742533411820,
0.085707919404388907742533411820, 0.37185068443447219278213697201, 0.55365197348122578372680248305, 0.867520158661744525773945397522, 1.14940378535561257656517445991, 1.31331939543842238212354345425, 1.50082046866480656820952361138, 1.72338282366061538859149420260, 1.75881668261124496464759877801, 1.93738935616731485086851826324, 1.99973945372021215876467136246, 2.43218289155926521795765659996, 2.56440454755023916427193009404, 2.61719772603265234114357990371, 2.74098675570014514149170913157, 2.88300001516468106532973696540, 3.08573186008268237773925384356, 3.23056591876677218344386182232, 3.25006493111005529755774307288, 3.31186567848027221997734856023, 3.54124683492880151760624993694, 3.58498268007406818316704038461, 3.67337520973268467818594404088, 3.85883202616619750356043760452, 3.88684416121596189171358957123
Plot not available for L-functions of degree greater than 10.
