L(s) = 1 | − 3-s + 2·5-s + 3·7-s − 9-s − 2·11-s − 2·13-s − 2·15-s + 7·17-s − 9·19-s − 3·21-s + 6·23-s + 3·25-s + 5·29-s + 5·31-s + 2·33-s + 6·35-s + 37-s + 2·39-s − 20·41-s − 2·43-s − 2·45-s − 6·47-s − 3·49-s − 7·51-s + 13·53-s − 4·55-s + 9·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.13·7-s − 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.516·15-s + 1.69·17-s − 2.06·19-s − 0.654·21-s + 1.25·23-s + 3/5·25-s + 0.928·29-s + 0.898·31-s + 0.348·33-s + 1.01·35-s + 0.164·37-s + 0.320·39-s − 3.12·41-s − 0.304·43-s − 0.298·45-s − 0.875·47-s − 3/7·49-s − 0.980·51-s + 1.78·53-s − 0.539·55-s + 1.19·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.336008608\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.336008608\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 54 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 120 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 158 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 26 T + 346 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.559202414867928651069858668655, −8.355162525083403676785383397804, −8.246827489604441903006458722531, −7.67117449863213213536049130570, −7.23585986510385669341080601304, −6.84169358848669726240826928162, −6.38446455160626141403482528478, −6.27886466657804272403601511331, −5.60444801949900884358286612260, −5.31162932849553428975449686695, −4.96553277235539371369580082976, −4.88035638560409393146406747047, −4.29590352062415944696367251610, −3.70017216051255956833919344856, −3.01720714862118923861259366411, −2.88726716973045666195765820789, −2.01232064853539611645643860034, −1.91029933032961674184459126658, −1.17526251650271530336063512894, −0.50005541082092645207911075490,
0.50005541082092645207911075490, 1.17526251650271530336063512894, 1.91029933032961674184459126658, 2.01232064853539611645643860034, 2.88726716973045666195765820789, 3.01720714862118923861259366411, 3.70017216051255956833919344856, 4.29590352062415944696367251610, 4.88035638560409393146406747047, 4.96553277235539371369580082976, 5.31162932849553428975449686695, 5.60444801949900884358286612260, 6.27886466657804272403601511331, 6.38446455160626141403482528478, 6.84169358848669726240826928162, 7.23585986510385669341080601304, 7.67117449863213213536049130570, 8.246827489604441903006458722531, 8.355162525083403676785383397804, 8.559202414867928651069858668655