| L(s) = 1 | − 4-s + 2·9-s + 16-s − 2·36-s + 2·49-s − 64-s + 3·81-s + 4·89-s − 4·101-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + ⋯ |
| L(s) = 1 | − 4-s + 2·9-s + 16-s − 2·36-s + 2·49-s − 64-s + 3·81-s + 4·89-s − 4·101-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.121119570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.121119570\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.550987724788604379431264379495, −9.493249164785248564425325113513, −8.947034852315499554544232547344, −8.685197224461549113258892412836, −8.109091838592266158893116795767, −7.69605804913797349617944069976, −7.43011918093119782203783332410, −7.07884851431177866454503796137, −6.41533775595560043029732049518, −6.29059528335442226075537437258, −5.53969064161163281918024913047, −5.06841425462184414006062935921, −4.87696887015863872538826880717, −4.21931866556954222613572583014, −3.90192539579700297486776828678, −3.70773392692290291612075294082, −2.84927817052814625843250387070, −2.22085192687463278883090077162, −1.48014408051926811139872589975, −0.948393339903121943391961748045,
0.948393339903121943391961748045, 1.48014408051926811139872589975, 2.22085192687463278883090077162, 2.84927817052814625843250387070, 3.70773392692290291612075294082, 3.90192539579700297486776828678, 4.21931866556954222613572583014, 4.87696887015863872538826880717, 5.06841425462184414006062935921, 5.53969064161163281918024913047, 6.29059528335442226075537437258, 6.41533775595560043029732049518, 7.07884851431177866454503796137, 7.43011918093119782203783332410, 7.69605804913797349617944069976, 8.109091838592266158893116795767, 8.685197224461549113258892412836, 8.947034852315499554544232547344, 9.493249164785248564425325113513, 9.550987724788604379431264379495
