| L(s) = 1 | − 2·7-s + 2·9-s − 4·17-s + 16·23-s + 10·25-s + 8·31-s + 20·41-s − 8·47-s + 3·49-s − 4·63-s + 12·73-s + 32·79-s − 5·81-s − 36·89-s − 4·97-s − 8·103-s + 12·113-s + 8·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2/3·9-s − 0.970·17-s + 3.33·23-s + 2·25-s + 1.43·31-s + 3.12·41-s − 1.16·47-s + 3/7·49-s − 0.503·63-s + 1.40·73-s + 3.60·79-s − 5/9·81-s − 3.81·89-s − 0.406·97-s − 0.788·103-s + 1.12·113-s + 0.733·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.949870662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.949870662\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.369710562972011561898019225554, −9.201349479857893891195869927889, −8.780258114064044893135732091323, −8.341934686997808659272342068663, −7.979389979744781503132578793131, −7.30901821941490448430624847922, −6.97463647373611669132256534549, −6.79075070783496549110793787172, −6.48848832724069200428806306213, −5.92586845921904679923455444827, −5.33387101468763420791424955233, −4.95340962889400583127230404474, −4.40818294931119028453807615181, −4.36689577416783662767066785268, −3.45603500129626999376678221219, −2.88895545738300268266069247958, −2.84803689659564379713923182865, −2.07029321070768850547863925729, −0.923510846156339039468053570658, −0.919586727046695428101921016490,
0.919586727046695428101921016490, 0.923510846156339039468053570658, 2.07029321070768850547863925729, 2.84803689659564379713923182865, 2.88895545738300268266069247958, 3.45603500129626999376678221219, 4.36689577416783662767066785268, 4.40818294931119028453807615181, 4.95340962889400583127230404474, 5.33387101468763420791424955233, 5.92586845921904679923455444827, 6.48848832724069200428806306213, 6.79075070783496549110793787172, 6.97463647373611669132256534549, 7.30901821941490448430624847922, 7.979389979744781503132578793131, 8.341934686997808659272342068663, 8.780258114064044893135732091323, 9.201349479857893891195869927889, 9.369710562972011561898019225554
