| L(s) = 1 | + 196·7-s + 480·9-s − 1.05e4·25-s + 4.97e4·37-s + 7.16e4·43-s − 4.80e3·49-s + 9.40e4·63-s − 2.78e4·67-s − 1.19e5·79-s + 1.71e5·81-s − 7.92e4·109-s − 9.76e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.99e5·169-s + 173-s − 2.07e6·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.97·9-s − 3.38·25-s + 5.97·37-s + 5.90·43-s − 2/7·49-s + 2.98·63-s − 0.758·67-s − 2.15·79-s + 2.90·81-s − 0.638·109-s − 0.606·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-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 + 2.15·169-s + 2.54e−6·173-s − 5.11·175-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(11.34272937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.34272937\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 - 160 p T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 p^{2} T + p^{5} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 5284 T^{2} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 48842 T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 399860 T^{2} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 424714 T^{2} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 1725952 T^{2} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 45370 p T^{2} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 11804662 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 57133886 T^{2} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12430 T + p^{5} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 222434938 T^{2} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 17900 T + p^{5} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 422333638 T^{2} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 216630502 T^{2} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 922094848 T^{2} + p^{10} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 1152129476 T^{2} + p^{10} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 104 p T + p^{5} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 3606974926 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2244248690 T^{2} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 29846 T + p^{5} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 385567936 T^{2} + p^{10} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 8595251314 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 11513107370 T^{2} + p^{10} T^{4} )^{2} \) |
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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
−7.58061224335581175912486680868, −7.37538698447973728352355343425, −7.36473842277479456148948527026, −6.67661696102995267689870019943, −6.45706827048048621815947043751, −6.17105215163422287855349903911, −5.93377938350816630784381000257, −5.69513123310007242290122373608, −5.59126603172939318230855079094, −5.18189982064113511383556578336, −4.48671149548700978440185654730, −4.47363179981894722041802438375, −4.32233670333602391353014730177, −4.19150163305189419756664352432, −3.99144436234907527941648441843, −3.55423150327486741338913091557, −2.93113639358012729663090249194, −2.51814583109382373341537638245, −2.34178184544612410232049098508, −2.20581367599265565236672358805, −1.55284272607574414352707483738, −1.36936861019948041321707148229, −1.05901992362117992858885451138, −0.72844990293485070638849693114, −0.38361110893300536112610978101,
0.38361110893300536112610978101, 0.72844990293485070638849693114, 1.05901992362117992858885451138, 1.36936861019948041321707148229, 1.55284272607574414352707483738, 2.20581367599265565236672358805, 2.34178184544612410232049098508, 2.51814583109382373341537638245, 2.93113639358012729663090249194, 3.55423150327486741338913091557, 3.99144436234907527941648441843, 4.19150163305189419756664352432, 4.32233670333602391353014730177, 4.47363179981894722041802438375, 4.48671149548700978440185654730, 5.18189982064113511383556578336, 5.59126603172939318230855079094, 5.69513123310007242290122373608, 5.93377938350816630784381000257, 6.17105215163422287855349903911, 6.45706827048048621815947043751, 6.67661696102995267689870019943, 7.36473842277479456148948527026, 7.37538698447973728352355343425, 7.58061224335581175912486680868
