| L(s) = 1 | − 9-s + 10·19-s + 8·29-s − 10·31-s − 20·41-s + 5·49-s − 20·59-s + 10·61-s − 20·71-s + 81-s − 32·89-s − 4·101-s + 10·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 10·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.29·19-s + 1.48·29-s − 1.79·31-s − 3.12·41-s + 5/7·49-s − 2.60·59-s + 1.28·61-s − 2.37·71-s + 1/9·81-s − 3.39·89-s − 0.398·101-s + 0.957·109-s − 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s − 0.764·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.346600143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.346600143\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
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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
−8.606938057412524202999607600907, −8.147241183398007594079077610277, −7.67477548776952458957959219757, −7.31491457990130331438229770146, −7.13286337879199020883355931321, −6.71455144011766870407364801856, −6.34142808508150924037838829011, −5.71731648523789680883129590211, −5.63779364263336864356753665397, −5.03888969184073267711721241618, −5.02972946519885782135700275423, −4.34739751737343956023405624561, −3.96055052278994043085248526918, −3.39543492773350939285457479261, −3.02840698547953707872664159336, −2.92978096099522322815181471280, −2.14295431639444958623724078918, −1.39241911126522342939522615265, −1.37607850092143067310430026516, −0.32081068454276407918212197672,
0.32081068454276407918212197672, 1.37607850092143067310430026516, 1.39241911126522342939522615265, 2.14295431639444958623724078918, 2.92978096099522322815181471280, 3.02840698547953707872664159336, 3.39543492773350939285457479261, 3.96055052278994043085248526918, 4.34739751737343956023405624561, 5.02972946519885782135700275423, 5.03888969184073267711721241618, 5.63779364263336864356753665397, 5.71731648523789680883129590211, 6.34142808508150924037838829011, 6.71455144011766870407364801856, 7.13286337879199020883355931321, 7.31491457990130331438229770146, 7.67477548776952458957959219757, 8.147241183398007594079077610277, 8.606938057412524202999607600907
