L(s) = 1 | + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s + 4·41-s + 2·53-s − 4·65-s − 2·73-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s + 4·41-s + 2·53-s − 4·65-s − 2·73-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.113065027\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.113065027\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_2^2$ | \( 1 + T^{4} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
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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
−9.122985483421329651393304358192, −9.054808555652830358511866688992, −8.451572014982720513571784719359, −7.947055717383514059880694882193, −7.53179085325822748549171207671, −7.28399161595257855237613723379, −6.87958905364385779778160339889, −6.53701648709589852388108456791, −5.82704323853452741440798708245, −5.70793900477979294766613176962, −5.33107753825434546417942632052, −5.21468584215457646832547000997, −4.37805761102425759304775548008, −4.27348818929452709643991685152, −3.36024568570201550238938503975, −2.93801762967853872878877683254, −2.43350896458390204821321676454, −2.28211909805043134554833880301, −1.46392526469955693743567413879, −1.00589148276618140205811909963,
1.00589148276618140205811909963, 1.46392526469955693743567413879, 2.28211909805043134554833880301, 2.43350896458390204821321676454, 2.93801762967853872878877683254, 3.36024568570201550238938503975, 4.27348818929452709643991685152, 4.37805761102425759304775548008, 5.21468584215457646832547000997, 5.33107753825434546417942632052, 5.70793900477979294766613176962, 5.82704323853452741440798708245, 6.53701648709589852388108456791, 6.87958905364385779778160339889, 7.28399161595257855237613723379, 7.53179085325822748549171207671, 7.947055717383514059880694882193, 8.451572014982720513571784719359, 9.054808555652830358511866688992, 9.122985483421329651393304358192