| L(s) = 1 | − 5-s − 6·7-s − 11-s + 10·13-s − 8·17-s + 5·19-s + 8·23-s − 4·25-s − 6·29-s + 2·31-s + 6·35-s − 3·37-s − 12·41-s − 14·43-s − 12·47-s + 13·49-s + 11·53-s + 55-s + 5·59-s − 20·61-s − 10·65-s − 7·67-s − 8·71-s + 73-s + 6·77-s − 8·79-s − 14·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 2.26·7-s − 0.301·11-s + 2.77·13-s − 1.94·17-s + 1.14·19-s + 1.66·23-s − 4/5·25-s − 1.11·29-s + 0.359·31-s + 1.01·35-s − 0.493·37-s − 1.87·41-s − 2.13·43-s − 1.75·47-s + 13/7·49-s + 1.51·53-s + 0.134·55-s + 0.650·59-s − 2.56·61-s − 1.24·65-s − 0.855·67-s − 0.949·71-s + 0.117·73-s + 0.683·77-s − 0.900·79-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07590273452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07590273452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + T - 7 T^{2} - 14 T^{3} - 68 T^{4} - 14 p T^{5} - 7 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 5 T - 5 T^{2} + 40 T^{3} + 64 T^{4} + 40 p T^{5} - 5 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 36 p T^{5} - 53 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 11 T - T^{2} - 176 T^{3} + 5662 T^{4} - 176 p T^{5} - p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5 T - 85 T^{2} + 40 T^{3} + 7144 T^{4} + 40 p T^{5} - 85 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 7 T - 83 T^{2} - 14 T^{3} + 9652 T^{4} - 14 p T^{5} - 83 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - T - 131 T^{2} + 14 T^{3} + 12022 T^{4} + 14 p T^{5} - 131 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T - 53 T^{2} - 328 T^{3} + 2392 T^{4} - 328 p T^{5} - 53 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 7 T + 164 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 94 T^{2} - 288 T^{3} + 5775 T^{4} - 288 p T^{5} - 94 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 25 T + 336 T^{2} - 25 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.88110030602600671525786931753, −7.67319735820688875431296218028, −7.18527564783267972656927301905, −7.06631439851383122304090628705, −6.92536146533550938377626533515, −6.78577233948623627012317825020, −6.27557007422861267016905538631, −6.23621073976120312592508574383, −6.07745464313649604759445013024, −5.87346250481402089294656252869, −5.33106496025951954773054567490, −5.24944313869593765526938279797, −4.77957998509779172005242921213, −4.65429732110921560121921876757, −4.24061181184462077621066371421, −3.77862539058331446078177189146, −3.60014165387891517542180192832, −3.39110734290764065388311607061, −3.32204951488817769804452977735, −2.85189444471231326359137951074, −2.64935304768218923267406787111, −1.90612425236773959998685281957, −1.52825652717293792338932886496, −1.21089466871120919642379493177, −0.095151750686273758143060150562,
0.095151750686273758143060150562, 1.21089466871120919642379493177, 1.52825652717293792338932886496, 1.90612425236773959998685281957, 2.64935304768218923267406787111, 2.85189444471231326359137951074, 3.32204951488817769804452977735, 3.39110734290764065388311607061, 3.60014165387891517542180192832, 3.77862539058331446078177189146, 4.24061181184462077621066371421, 4.65429732110921560121921876757, 4.77957998509779172005242921213, 5.24944313869593765526938279797, 5.33106496025951954773054567490, 5.87346250481402089294656252869, 6.07745464313649604759445013024, 6.23621073976120312592508574383, 6.27557007422861267016905538631, 6.78577233948623627012317825020, 6.92536146533550938377626533515, 7.06631439851383122304090628705, 7.18527564783267972656927301905, 7.67319735820688875431296218028, 7.88110030602600671525786931753
