| L(s) = 1 | + 4-s − 2·7-s − 4·13-s + 4·16-s + 20·19-s + 7·25-s − 2·28-s − 10·31-s − 28·37-s + 8·43-s + 49-s − 4·52-s − 16·61-s + 11·64-s − 28·67-s − 16·73-s + 20·76-s − 16·79-s + 8·91-s + 8·97-s + 7·100-s − 10·103-s − 28·109-s − 8·112-s + 19·121-s − 10·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 1.10·13-s + 16-s + 4.58·19-s + 7/5·25-s − 0.377·28-s − 1.79·31-s − 4.60·37-s + 1.21·43-s + 1/7·49-s − 0.554·52-s − 2.04·61-s + 11/8·64-s − 3.42·67-s − 1.87·73-s + 2.29·76-s − 1.80·79-s + 0.838·91-s + 0.812·97-s + 7/10·100-s − 0.985·103-s − 2.68·109-s − 0.755·112-s + 1.72·121-s − 0.898·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.685304307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.685304307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 19 T^{2} + 240 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 43 T^{2} + 1320 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 50 T^{2} + 1659 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 115 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 58 T^{2} - 3525 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 4 T - 81 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.76740824301625058812568832826, −7.31916807953055092475224895583, −7.18006444722374662995972729642, −7.14273043848672797492948048751, −7.10419160932067670502991088999, −6.70300913631473854493731441770, −6.24861291395846829319973736777, −6.03345830149060292034882977508, −5.71456967415541527279950685135, −5.47406321642738930123320060712, −5.35580065176301771723120314484, −5.10334021506319111420548888182, −5.00053325509720938335022166426, −4.58461464960830988142800607231, −4.14757489423840472322823611918, −3.76067940591664971832802249672, −3.46622106553540447445582025502, −3.33466127005194170598918501691, −2.92541520826229938449365536738, −2.83621259073170263146515778201, −2.66551733316875358270433165724, −1.62030256866595147010950196573, −1.54497310497130911948835887036, −1.37114617351508921125778695066, −0.39620014345162603517039108289,
0.39620014345162603517039108289, 1.37114617351508921125778695066, 1.54497310497130911948835887036, 1.62030256866595147010950196573, 2.66551733316875358270433165724, 2.83621259073170263146515778201, 2.92541520826229938449365536738, 3.33466127005194170598918501691, 3.46622106553540447445582025502, 3.76067940591664971832802249672, 4.14757489423840472322823611918, 4.58461464960830988142800607231, 5.00053325509720938335022166426, 5.10334021506319111420548888182, 5.35580065176301771723120314484, 5.47406321642738930123320060712, 5.71456967415541527279950685135, 6.03345830149060292034882977508, 6.24861291395846829319973736777, 6.70300913631473854493731441770, 7.10419160932067670502991088999, 7.14273043848672797492948048751, 7.18006444722374662995972729642, 7.31916807953055092475224895583, 7.76740824301625058812568832826
