L(s) = 1 | + 3-s + 3·7-s + 3·21-s + 3·23-s − 27-s − 29-s − 41-s − 3·47-s + 5·49-s − 61-s + 3·67-s + 3·69-s − 81-s − 3·83-s − 87-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 123-s + 127-s + 131-s + 137-s + 139-s − 3·141-s + 5·147-s + 149-s + ⋯ |
L(s) = 1 | + 3-s + 3·7-s + 3·21-s + 3·23-s − 27-s − 29-s − 41-s − 3·47-s + 5·49-s − 61-s + 3·67-s + 3·69-s − 81-s − 3·83-s − 87-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 123-s + 127-s + 131-s + 137-s + 139-s − 3·141-s + 5·147-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.190132412\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.190132412\) |
\(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−8.774233293235374318073716510000, −8.413552785979424476797980298620, −8.223318241315047015748702234177, −7.939884712043278586036290269757, −7.64955022906691997211082746268, −7.19968733023872853066613802129, −6.74106956231082083138956396413, −6.60735342854946417288903520780, −5.51852554434213728838964364638, −5.50827504382956451802312524405, −5.08713894358954790474357448754, −4.77111005575150577678909005913, −4.45266255159392665606585548180, −3.87043298703629266043180909715, −3.40271583100355970322259024002, −2.92890576232569559654620344112, −2.51315371053266114050122915054, −1.79692073878321632397685117391, −1.63846770763424277429347827945, −1.10428701198058591233431240327,
1.10428701198058591233431240327, 1.63846770763424277429347827945, 1.79692073878321632397685117391, 2.51315371053266114050122915054, 2.92890576232569559654620344112, 3.40271583100355970322259024002, 3.87043298703629266043180909715, 4.45266255159392665606585548180, 4.77111005575150577678909005913, 5.08713894358954790474357448754, 5.50827504382956451802312524405, 5.51852554434213728838964364638, 6.60735342854946417288903520780, 6.74106956231082083138956396413, 7.19968733023872853066613802129, 7.64955022906691997211082746268, 7.939884712043278586036290269757, 8.223318241315047015748702234177, 8.413552785979424476797980298620, 8.774233293235374318073716510000