| L(s) = 1 | + 16·2-s + 136·4-s + 816·8-s + 3.87e3·16-s − 8·23-s − 3·25-s + 1.55e4·32-s − 128·46-s + 49-s − 48·50-s + 16·53-s + 5.42e4·64-s − 8·79-s − 1.08e3·92-s + 16·98-s − 408·100-s + 256·106-s + 12·107-s − 4·109-s + 60·113-s + 76·121-s + 127-s + 1.70e5·128-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 11.3·2-s + 68·4-s + 288.·8-s + 969·16-s − 1.66·23-s − 3/5·25-s + 2.74e3·32-s − 18.8·46-s + 1/7·49-s − 6.78·50-s + 2.19·53-s + 6.78e3·64-s − 0.900·79-s − 113.·92-s + 1.61·98-s − 40.7·100-s + 24.8·106-s + 1.16·107-s − 0.383·109-s + 5.64·113-s + 6.90·121-s + 0.0887·127-s + 1.50e4·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{48} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{48} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(106013.1882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(106013.1882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{16} \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 3 T^{2} - 2 p T^{4} + 33 T^{6} + 474 T^{8} + 33 p^{2} T^{10} - 2 p^{5} T^{12} + 3 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - T^{2} - 22 T^{4} - 15 T^{6} + 554 T^{8} - 15 p^{2} T^{10} - 22 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \) |
| good | 11 | \( ( 1 - 38 T^{2} + 83 p T^{4} - 15590 T^{6} + 194008 T^{8} - 15590 p^{2} T^{10} + 83 p^{5} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 53 T^{2} + 1421 T^{4} + 26118 T^{6} + 376130 T^{8} + 26118 p^{2} T^{10} + 1421 p^{4} T^{12} + 53 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 - 58 T^{2} + 2009 T^{4} - 49530 T^{6} + 951284 T^{8} - 49530 p^{2} T^{10} + 2009 p^{4} T^{12} - 58 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 81 T^{2} + 3350 T^{4} - 98331 T^{6} + 2169690 T^{8} - 98331 p^{2} T^{10} + 3350 p^{4} T^{12} - 81 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 + 2 T + 33 T^{2} + 820 T^{4} + 33 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 134 T^{2} + 9253 T^{4} - 425582 T^{6} + 14258284 T^{8} - 425582 p^{2} T^{10} + 9253 p^{4} T^{12} - 134 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 159 T^{2} + 12581 T^{4} - 641190 T^{6} + 23260110 T^{8} - 641190 p^{2} T^{10} + 12581 p^{4} T^{12} - 159 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - 196 T^{2} + 19640 T^{4} - 1252476 T^{6} + 55246958 T^{8} - 1252476 p^{2} T^{10} + 19640 p^{4} T^{12} - 196 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 + 159 T^{2} + 13778 T^{4} + 840417 T^{6} + 39217914 T^{8} + 840417 p^{2} T^{10} + 13778 p^{4} T^{12} + 159 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 295 T^{2} + 39917 T^{4} - 3224142 T^{6} + 169742774 T^{8} - 3224142 p^{2} T^{10} + 39917 p^{4} T^{12} - 295 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 + 15 T^{2} + 2378 T^{4} - 71535 T^{6} + 4137834 T^{8} - 71535 p^{2} T^{10} + 2378 p^{4} T^{12} + 15 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 4 T + 72 T^{2} + 96 T^{3} + 3025 T^{4} + 96 p T^{5} + 72 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 198 T^{2} + 24269 T^{4} + 2110662 T^{6} + 143664636 T^{8} + 2110662 p^{2} T^{10} + 24269 p^{4} T^{12} + 198 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - p T^{2} )^{16} \) |
| 67 | \( ( 1 - 79 T^{2} + 5069 T^{4} - 215790 T^{6} - 4579210 T^{8} - 215790 p^{2} T^{10} + 5069 p^{4} T^{12} - 79 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 - 386 T^{2} + 71989 T^{4} - 8520110 T^{6} + 710279188 T^{8} - 8520110 p^{2} T^{10} + 71989 p^{4} T^{12} - 386 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 + 306 T^{2} + 55085 T^{4} + 6477858 T^{6} + 556796076 T^{8} + 6477858 p^{2} T^{10} + 55085 p^{4} T^{12} + 306 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 + 2 T + 110 T^{2} - 510 T^{3} + 3746 T^{4} - 510 p T^{5} + 110 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 586 T^{2} + 154913 T^{4} - 24245394 T^{6} + 2465561876 T^{8} - 24245394 p^{2} T^{10} + 154913 p^{4} T^{12} - 586 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 189 T^{2} + 9461 T^{4} + 745146 T^{6} + 124208310 T^{8} + 745146 p^{2} T^{10} + 9461 p^{4} T^{12} + 189 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 + 507 T^{2} + 129614 T^{4} + 21362481 T^{6} + 2454936522 T^{8} + 21362481 p^{2} T^{10} + 129614 p^{4} T^{12} + 507 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.45348567388881946147579127601, −2.39632416189927448060925107404, −2.31854896004171039394243049173, −2.31292047908962350824753033975, −2.18460068660082757347776143645, −2.12307369844956200007287789761, −2.06681188741962699057463289203, −2.02127655130028249478964148942, −1.81895228971392256150144251193, −1.75841617120847425593272844851, −1.74028460981612736253316780543, −1.72196798143336353161542931334, −1.67414188572632956270359753695, −1.61751410938646875628105753978, −1.47019149003000871838632787430, −1.37022295777218948746516518401, −1.24957491503975616051541714211, −1.15873779453865108135958287472, −0.986585148320075131378771768246, −0.912839357051618389832099713394, −0.792942038304288899557272502250, −0.68566350654067102034096722400, −0.60351879379757752957507975466, −0.47894471554261800151443168239, −0.11432963163876662064956676855,
0.11432963163876662064956676855, 0.47894471554261800151443168239, 0.60351879379757752957507975466, 0.68566350654067102034096722400, 0.792942038304288899557272502250, 0.912839357051618389832099713394, 0.986585148320075131378771768246, 1.15873779453865108135958287472, 1.24957491503975616051541714211, 1.37022295777218948746516518401, 1.47019149003000871838632787430, 1.61751410938646875628105753978, 1.67414188572632956270359753695, 1.72196798143336353161542931334, 1.74028460981612736253316780543, 1.75841617120847425593272844851, 1.81895228971392256150144251193, 2.02127655130028249478964148942, 2.06681188741962699057463289203, 2.12307369844956200007287789761, 2.18460068660082757347776143645, 2.31292047908962350824753033975, 2.31854896004171039394243049173, 2.39632416189927448060925107404, 2.45348567388881946147579127601
Plot not available for L-functions of degree greater than 10.
