L(s) = 1 | + 2·3-s + 3·9-s − 8·25-s + 4·27-s − 8·31-s − 8·37-s − 24·47-s − 20·53-s − 16·75-s + 5·81-s + 8·83-s − 16·93-s − 24·103-s + 8·109-s − 16·111-s − 4·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s − 48·141-s + 149-s + 151-s + 157-s − 40·159-s + 163-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 8/5·25-s + 0.769·27-s − 1.43·31-s − 1.31·37-s − 3.50·47-s − 2.74·53-s − 1.84·75-s + 5/9·81-s + 0.878·83-s − 1.65·93-s − 2.36·103-s + 0.766·109-s − 1.51·111-s − 0.376·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.04·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 3.17·159-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−7.47387458896252166902402646729, −7.39961299145506272372634150237, −6.82273205312770361074092063355, −6.67614092340468741133135433208, −6.13254126799236570809664307564, −5.98530660742761897174697736855, −5.32051910888537439176944341085, −5.15722395037922820715191160760, −4.63526592857681755901962206415, −4.45486381885996857575382657400, −3.76137653143114479779026594425, −3.63716339073710508564694694954, −3.17483802462248879729345800131, −3.07622635213107252059363881560, −2.19219440194281609499012603618, −2.10384094357741018328998779064, −1.52168097207756385183619815163, −1.32335561967512417535557346485, 0, 0,
1.32335561967512417535557346485, 1.52168097207756385183619815163, 2.10384094357741018328998779064, 2.19219440194281609499012603618, 3.07622635213107252059363881560, 3.17483802462248879729345800131, 3.63716339073710508564694694954, 3.76137653143114479779026594425, 4.45486381885996857575382657400, 4.63526592857681755901962206415, 5.15722395037922820715191160760, 5.32051910888537439176944341085, 5.98530660742761897174697736855, 6.13254126799236570809664307564, 6.67614092340468741133135433208, 6.82273205312770361074092063355, 7.39961299145506272372634150237, 7.47387458896252166902402646729