| L(s) = 1 | + 128·4-s − 112·7-s + 8.16e3·16-s − 7.81e3·25-s − 1.43e4·28-s − 3.43e4·37-s + 3.24e4·43-s − 4.67e4·49-s + 3.87e5·64-s + 1.52e5·67-s − 9.45e4·79-s − 1.00e6·100-s − 6.41e5·109-s − 9.13e5·112-s + 1.12e6·121-s + 127-s + 131-s + 137-s + 139-s − 4.39e6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.61e6·169-s + 4.14e6·172-s + ⋯ |
| L(s) = 1 | + 4·4-s − 0.863·7-s + 7.96·16-s − 2.50·25-s − 3.45·28-s − 4.11·37-s + 2.67·43-s − 2.78·49-s + 11.8·64-s + 4.15·67-s − 1.70·79-s − 10.0·100-s − 5.16·109-s − 6.88·112-s + 7.01·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 16.4·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4.35·169-s + 10.6·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(10.82514849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.82514849\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 + 8 p T + 4014 p T^{2} + 8 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| good | 2 | \( ( 1 - p^{6} T^{2} + 129 p^{4} T^{4} - p^{16} T^{6} + p^{20} T^{8} )^{2} \) |
| 5 | \( ( 1 + 3908 T^{2} + 23248566 T^{4} + 3908 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 11 | \( ( 1 - 564976 T^{2} + 131629015794 T^{4} - 564976 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 13 | \( ( 1 - 809044 T^{2} + 351572118390 T^{4} - 809044 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 17 | \( ( 1 + 1883444 T^{2} + 1329835154982 T^{4} + 1883444 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 19 | \( ( 1 - 190372 p T^{2} + 10651698082806 T^{4} - 190372 p^{11} T^{6} + p^{20} T^{8} )^{2} \) |
| 23 | \( ( 1 - 4853968 T^{2} + 70775754302754 T^{4} - 4853968 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 29 | \( ( 1 - 61078432 T^{2} + 1759165582775058 T^{4} - 61078432 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 31 | \( ( 1 - 23985916 T^{2} + 1685938631161734 T^{4} - 23985916 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 37 | \( ( 1 + 8576 T + 87352506 T^{2} + 8576 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 41 | \( ( 1 + 13154708 T^{2} - 843745561706682 T^{4} + 13154708 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 43 | \( ( 1 - 8104 T + 215741802 T^{2} - 8104 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 47 | \( ( 1 + 102260732 T^{2} + 1855613853315738 p T^{4} + 102260732 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 53 | \( ( 1 - 334196128 T^{2} + 373722291304309746 T^{4} - 334196128 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 59 | \( ( 1 + 1446882860 T^{2} + 1156390163071869654 T^{4} + 1446882860 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 - 2484649012 T^{2} + 2966879913883883766 T^{4} - 2484649012 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 67 | \( ( 1 - 38152 T + 1428083382 T^{2} - 38152 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 71 | \( ( 1 - 1243284688 T^{2} - 1234162537544764062 T^{4} - 1243284688 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4781796484 T^{2} + 14254537270266844710 T^{4} - 4781796484 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 79 | \( ( 1 + 23648 T + 5558769246 T^{2} + 23648 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 83 | \( ( 1 + 10148193356 T^{2} + 54914978401780671414 T^{4} + 10148193356 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 89 | \( ( 1 + 9743291348 T^{2} + 65866481293586129286 T^{4} + 9743291348 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 97 | \( ( 1 - 23990990692 T^{2} + \)\(26\!\cdots\!46\)\( T^{4} - 23990990692 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
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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
−5.99987596358565077100681909887, −5.87532114536495901278604646725, −5.75245659399435339757384192247, −5.68968403530144451892722145014, −5.38061575382163888234741847338, −5.17547361120411653801869902733, −4.97211807354375927014797695442, −4.72336136707475224002748572135, −4.34366987904117239093636521185, −3.98713538531563522194279044680, −3.89166789721544711246690398887, −3.71077479789247721467898710194, −3.41754901480310806891735600514, −3.12640913640859673578437851357, −3.06793444095434570633307802266, −2.88185664015365490019875297192, −2.35918717020203076723639864426, −2.19434043480898196305388073930, −2.18852073557731460977950317737, −1.83298449805950142665771517312, −1.71242502757748583163647810458, −1.43857374598547876198375588578, −0.978752639084975108683852956534, −0.50348456982078222430521305368, −0.26681910474026482952579214774,
0.26681910474026482952579214774, 0.50348456982078222430521305368, 0.978752639084975108683852956534, 1.43857374598547876198375588578, 1.71242502757748583163647810458, 1.83298449805950142665771517312, 2.18852073557731460977950317737, 2.19434043480898196305388073930, 2.35918717020203076723639864426, 2.88185664015365490019875297192, 3.06793444095434570633307802266, 3.12640913640859673578437851357, 3.41754901480310806891735600514, 3.71077479789247721467898710194, 3.89166789721544711246690398887, 3.98713538531563522194279044680, 4.34366987904117239093636521185, 4.72336136707475224002748572135, 4.97211807354375927014797695442, 5.17547361120411653801869902733, 5.38061575382163888234741847338, 5.68968403530144451892722145014, 5.75245659399435339757384192247, 5.87532114536495901278604646725, 5.99987596358565077100681909887
Plot not available for L-functions of degree greater than 10.
