| L(s) = 1 | + 2·2-s + 4-s + 2·5-s + 2·7-s − 2·8-s + 4·10-s + 4·11-s + 4·14-s − 4·16-s − 8·17-s + 20·19-s + 2·20-s + 8·22-s − 2·23-s + 5·25-s + 2·28-s − 4·29-s − 12·31-s − 2·32-s − 16·34-s + 4·35-s + 8·37-s + 40·38-s − 4·40-s − 4·43-s + 4·44-s − 4·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.894·5-s + 0.755·7-s − 0.707·8-s + 1.26·10-s + 1.20·11-s + 1.06·14-s − 16-s − 1.94·17-s + 4.58·19-s + 0.447·20-s + 1.70·22-s − 0.417·23-s + 25-s + 0.377·28-s − 0.742·29-s − 2.15·31-s − 0.353·32-s − 2.74·34-s + 0.676·35-s + 1.31·37-s + 6.48·38-s − 0.632·40-s − 0.609·43-s + 0.603·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.307772300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.307772300\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 p T^{3} - 4 p T^{4} + 2 p^{2} T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T - 22 T^{2} - 80 T^{3} + 139 T^{4} - 80 p T^{5} - 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 80 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 2 T^{2} - 2205 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 1350 p T^{5} + 127 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 640 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 90 p T^{5} - 107 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 80 p T^{5} - 158 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.329511368106140275767813296213, −7.73367092251083696214711874071, −7.59217796900276123980170463359, −7.39864112329380115867568751883, −7.31063589104178823021623345883, −7.07150880346680772691937068749, −6.46961497929798402872694088916, −6.46019671878303665795526071095, −6.00097999082114029471775925221, −5.96830505884630385383010518726, −5.56412698012461024194042146986, −5.32617692616098358284295301354, −4.97303954481632471836363213763, −4.96646380610646230688410994537, −4.79790878974330729867877771018, −4.05339457385563941244863266042, −3.97174183351867101845986745049, −3.81865948812541205442946576738, −3.36279797727910818697042417223, −3.02968964935376171483801971320, −2.69954442091632307699708298370, −2.30096668731260982282408203411, −1.73462855171427635585311649090, −1.44334279050637367970325691865, −0.850324719538603196320751462879,
0.850324719538603196320751462879, 1.44334279050637367970325691865, 1.73462855171427635585311649090, 2.30096668731260982282408203411, 2.69954442091632307699708298370, 3.02968964935376171483801971320, 3.36279797727910818697042417223, 3.81865948812541205442946576738, 3.97174183351867101845986745049, 4.05339457385563941244863266042, 4.79790878974330729867877771018, 4.96646380610646230688410994537, 4.97303954481632471836363213763, 5.32617692616098358284295301354, 5.56412698012461024194042146986, 5.96830505884630385383010518726, 6.00097999082114029471775925221, 6.46019671878303665795526071095, 6.46961497929798402872694088916, 7.07150880346680772691937068749, 7.31063589104178823021623345883, 7.39864112329380115867568751883, 7.59217796900276123980170463359, 7.73367092251083696214711874071, 8.329511368106140275767813296213
