L(s) = 1 | + 2·3-s − 6·7-s + 2·9-s − 12·21-s − 2·23-s + 6·27-s + 12·29-s + 18·43-s − 14·47-s + 18·49-s − 12·63-s + 6·67-s − 4·69-s + 11·81-s − 22·83-s + 24·87-s + 12·89-s + 36·101-s − 18·103-s − 26·107-s − 22·121-s + 127-s + 36·129-s + 131-s + 137-s + 139-s − 28·141-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 2.26·7-s + 2/3·9-s − 2.61·21-s − 0.417·23-s + 1.15·27-s + 2.22·29-s + 2.74·43-s − 2.04·47-s + 18/7·49-s − 1.51·63-s + 0.733·67-s − 0.481·69-s + 11/9·81-s − 2.41·83-s + 2.57·87-s + 1.27·89-s + 3.58·101-s − 1.77·103-s − 2.51·107-s − 2·121-s + 0.0887·127-s + 3.16·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.35·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.159837556\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.159837556\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−9.523174034574868182466573055618, −9.304823470964272597343486247923, −8.758782447945612220320472704082, −8.595764309405587875172002727300, −8.021540869605553858873213033890, −7.74160705500498821782913137461, −7.16893796246024063340741053583, −6.68736585223604210366941734268, −6.50034022746275709810121668260, −6.18064424302070211711800588510, −5.60639137499272190758566589253, −5.04275692750156548948966532661, −4.34930608766521956900303016155, −4.12172957298135026813721255034, −3.37088856004627874714950864702, −3.20539849279450466192689145743, −2.57845352367891643340287237364, −2.49969222430741093346860044221, −1.36846386869001257293072000015, −0.54487369704968792806689364069,
0.54487369704968792806689364069, 1.36846386869001257293072000015, 2.49969222430741093346860044221, 2.57845352367891643340287237364, 3.20539849279450466192689145743, 3.37088856004627874714950864702, 4.12172957298135026813721255034, 4.34930608766521956900303016155, 5.04275692750156548948966532661, 5.60639137499272190758566589253, 6.18064424302070211711800588510, 6.50034022746275709810121668260, 6.68736585223604210366941734268, 7.16893796246024063340741053583, 7.74160705500498821782913137461, 8.021540869605553858873213033890, 8.595764309405587875172002727300, 8.758782447945612220320472704082, 9.304823470964272597343486247923, 9.523174034574868182466573055618